Polynomials — Class 10 Maths Notes

Topics covered in Polynomials

1. 1.Introduction of Polynomials

   What is a Polynomial?
   

   Imagine you are playing with building blocks. Each block can have a number on it and some blocks can be bigger or smaller than others. A polynomial is like a tower you build using these blocks. In math, these "blocks" are numbers and variables (like x, y, etc.) combined in different ways.

   Parts of a Polynomial:

   1. Terms: These are the building blocks. Each term is a combination of numbers and variables. For example, in 2x² + 3x + 5, there are three terms: 2x², 3x, and 5.
   2. Coefficients: These are the numbers in front of the variables. In 2x², 2 is the coefficient.
   3. Variables: These are letters that represent unknown numbers. In 2x², x is the variable.
   4. Exponents: These tell you how many times to multiply the variable by itself. In 2x², the exponent is 2, meaning x is multiplied by itself once (x*x).

   Types of Polynomials:

   1. Monomial: Just one term (like 5x).
   2. Binomial: Two terms (like x + 2).
   3. Trinomial: Three terms (like x² - 3x + 2).

   How to Use Polynomials:

   You can add, subtract, multiply, and sometimes divide polynomials. It's like rearranging or combining your building blocks to make new shapes.

   Where Are Polynomials Used?

   1. Real Life: Polynomials help in calculating areas, predicting growth (like population or interest), and in science for representing relationships between variables.
   2. Careers: Engineers, economists, scientists, and many other professionals use polynomials to solve problems.

   Activity:

   Try creating your own polynomial! Pick some numbers and variables, and put them together like building blocks. For example, choose a number like 4, a variable like x, and make a term like 4x. Then, add another term like + 5. Your polynomial is 4x + 5!

2. 2.Geometrical Meaning of the Zeroes of a Polynomial

   The zeroes of a polynomial have an interesting geometrical meaning, especially when we graph the polynomial on a coordinate plane. Let's break it down in a simple way:

   What are Zeroes of a Polynomial?

   The zeroes of a polynomial are the values of the variable (like x) that make the polynomial equal to zero. For example, in the polynomial x² - 4, the zeroes are x = 2 and x = -2 because when you put 2 or -2 in place of x, the polynomial equals zero.

   Geometrical Meaning:

   When you graph a polynomial, the zeroes are the points where the graph crosses or touches the x-axis. Here's what this means:

   1. Graphing a Polynomial: Imagine drawing a line or a curve on graph paper. This line or curve represents the polynomial.
   2. X-axis Crossings: The points where this line or curve crosses the x-axis (the horizontal line) are the zeroes of the polynomial.
   3. Touching the X-axis: Sometimes, the graph touches the x-axis but doesn't cross it. These points are also zeroes.

   Example:

   Take the polynomial x² - 4 again. If you graph it, you'll see that the curve touches the x-axis at x = 2 and x = -2. These points are the zeroes.

   Why It's Important:

   Understanding the zeroes geometrically helps in visualizing how polynomials behave. This is useful in many fields like physics, engineering, and economics, where graphing is a common tool to understand relationships and predict outcomes.

   Activity:

   Try graphing a simple polynomial like x² - 4. Use graph paper or a digital graphing tool. Notice where the graph crosses the x-axis and find the zeroes.

3. 3.Graphical method to find zeroes.

   To find the zeroes of a polynomial graphically, follow these steps:

   1. Plot the Polynomial: First, you need to create the graph of the polynomial p(x). This is typically done by calculating and plotting several points that satisfy the equation =y=p(x) and then drawing a smooth curve through these points.

   2. Identify the X-Axis: On a graph, the x-axis is the horizontal line. It is the line where the value of y is zero.

   3. Look for Intersections: The points where the curve crosses the x-axis are where =0y=0. These are the zeroes or roots of the polynomial. Each intersection point's x-coordinate is a zero of the polynomial.

   4. Count Touches and Crosses: A curve may touch the x-axis at a point and bounce off; this is still considered a zero, typically of even multiplicity (like (−)2(x−a)2). A curve that crosses the x-axis has a zero of odd multiplicity.

   5. Note the Zeroes: Write down the x-coordinate of each point where the curve touches or crosses the x-axis. These are the values of x for which =0p(x)=0.

   For example, if you have a polynomial =2−4p(x)=x2−4, the graph will be a parabola opening upwards. The points where this parabola crosses the x-axis are =−2x=−2 and =2x=2. Thus, the zeroes of the polynomial p(x) are -2 and 2.

   Let's apply these steps to find the zeroes for the polynomial graphs you've provided:

   • Top-Left Graph: No crossing or touching point, so no zeroes.
   • Top-Center Graph: One touching point, indicating a repeated zero (even multiplicity), so 1 zero.
   • Top-Right Graph: One crossing point, so 1 zero.
   • Bottom-Left Graph: One crossing point, so 1 zero.
   • Bottom-Center Graph: Three crossing points, so 3 zeroes.
   • Bottom-Right Graph: Four crossing points, so 4 zeroes.

   Remember, the exact value of the zeroes can be found by reading the x-coordinate of the points where the graph intersects the x-axis. If you need a precise value and the graph is not crossing at a labeled point on the axis, you may need to use other methods such as algebraic solving or numerical approximation to find the exact zeroes.

4. 4.The graphs of y = p(x) are (given in Fig.) below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

   
   
   

   To determine the number of zeroes of the polynomial p(x) for each graph using the graphical method, we will count how many times the graph crosses or touches the x-axis. Here's the analysis for each graph:

   1. Top-Left Graph: The graph is a horizontal line above the x-axis and does not intersect the x-axis at any point. Therefore, p(x) has 0 zeroes.

   2. Top-Center Graph: The graph touches the x-axis at one point but does not cross it. This is typically indicative of a double root (a zero of even multiplicity), but in terms of the number of distinct real zeroes, there is 1 zero.

   3. Top-Right Graph: The graph crosses the x-axis at one point, which means p(x) has 1 zero.

   4. Bottom-Left Graph: The graph intersects the x-axis at one point. Thus, p(x) has 1 zero.

   5. Bottom-Center Graph: The graph crosses the x-axis three times. Therefore, p(x) has 3 zeroes.

   6. Bottom-Right Graph: The graph intersects the x-axis four times. Consequently, p(x) has 4 zeroes.

   In summary, the number of zeroes for each graph is as follows:

   • Top-Left Graph: 0 zeroes
   • Top-Center Graph: 1 zero
   • Top-Right Graph: 1 zero
   • Bottom-Left Graph: 1 zero
   • Bottom-Center Graph: 3 zeroes
   • Bottom-Right Graph: 4 zeroes

   These zeroes are the x-values where the graph meets the x-axis, and they represent the solutions to the equation =0p(x)=0.

5. 5.Relationship between Zeroes and Coefficients of a Polynomial

   The relationship between the zeroes and coefficients of a polynomial is a fundamental concept in algebra. For a polynomial in standard form, which looks like +−1−1+⋯+1+0an​xn+an−1​xn−1+⋯+a1​x+a0​, where an​ is the leading coefficient and 0a0​ is the constant term, certain patterns arise that connect the coefficients to the zeroes of the polynomial.

   For a quadratic polynomial, for example, 2++ax2+bx+c, the sum and product of its zeroes (let's call them α and β) are related to the coefficients in the following ways:

   1. Sum of Zeroes: The sum of the zeroes +α+β is equal to the negative coefficient of x divided by the coefficient of 2x2, which gives us +=−/α+β=−b/a.

   2. Product of Zeroes: The product of the zeroes ⋅α⋅β is equal to the constant term divided by the coefficient of 2x2, resulting in ⋅=/α⋅β=c/a.

   For polynomials of higher degrees, similar relationships can be derived from the polynomial's factored form. For a cubic polynomial 3+2++ax3+bx2+cx+d, if the roots are,,α,β, and γ, then:

   1. Sum of Zeroes: ++=−/α+β+γ=−b/a.
   2. Sum of Product of Zeroes Taken Two at a Time: ++=/αβ+βγ+γα=c/a.
   3. Product of Zeroes: ⋅⋅=−/α⋅β⋅γ=−d/a.

   These relationships are a direct result of the polynomial being expressed in its factored form, (−)(−)(−)a(x−α)(x−β)(x−γ)... and so on for polynomials of even higher degrees.

6. 6.Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

   (i) x2−2x−8
   
   
   

   To factorize this quadratic, we need two numbers that multiply to give -8 (the constant term) and add up to -2 (the coefficient of x). These two numbers are -4 and +2.

   So we can write the polynomial as: 2−4+2−8x2−4x+2x−8

   Grouping the terms: (2−4)+(2−8)(x2−4x)+(2x−8)

   Factor out the common factors in each group: (−4)+2(−4)x(x−4)+2(x−4)

   Now factor out (−4)(x−4): (+2)(−4)(x+2)(x−4)

   The zeroes are the solutions to (+2)=0(x+2)=0 and (−4)=0(x−4)=0, which are =−2x=−2 and =4x=4.

   Verification: The sum of the zeroes (−2)+(4)=2(−2)+(4)=2 matches −−21−1−2​ and the product (−2)×(4)=−8(−2)×(4)=−8 matches the constant term -8.

   (ii) 42−4+14s2−4s+1

   This is a perfect square trinomial, which means it can be factored into the square of a binomial. It's the square of (2−1)(2s−1): (2−1)2(2s−1)2

   So the only zero is =12s=21​, which is a repeated root.

   Verification: The sum of the zeroes 12+12=121​+21​=1 matches −−44−4−4​, and the product 12×12=1421​×21​=41​ matches 1441​.

   (iii) 62−7−36x2−7x−3

   We are looking for two numbers that multiply to 6×−3=−186×−3=−18 and add up to -7. These numbers are -9 and +2.

   So we can write: 62−9+2−36x2−9x+2x−3

   Grouping: 3(2−3)+1(2−3)3x(2x−3)+1(2x−3)

   Factoring out (2−3)(2x−3): (3+1)(2−3)(3x+1)(2x−3)

   The zeroes are =−13x=−31​ and =32x=23​.

   Verification: The sum of the zeroes −13+32=76−31​+23​=67​ matches −−76−6−7​, and the product −13×32=−12−31​×23​=−21​ matches −366−3​.

   (iv) 42+84u2+8u

   Factor out the common factor of 4u: 4(+2)4u(u+2)

   The zeroes are =0u=0 and =−2u=−2.

   Verification: The sum of the zeroes 0+(−2)=−20+(−2)=−2 matches −84−48​, and the product 0×(−2)=00×(−2)=0 matches 0440​.

   (v) 2−15t2−15

   This polynomial needs two numbers that multiply to -15 and add up to 0. Since there is no x-term, we simply find the square roots of 15: (+15)(−15)(t+15​)(t−15​)

   The zeroes are =15t=15​ and =−15t=−15​.

   Verification: The sum of the zeroes 15+(−15)=015​+(−15​)=0 matches −01−10​, and the product 15×(−15)=−1515​×(−15​)=−15 matches −1511−15​.

   (vi) 32−−43x2−x−4

   We need two numbers that multiply to 3×−4=−123×−4=−12 and add up to -1. These numbers are -4 and +3.

   So we can write: 32−4+3−43x2−4x+3x−4

   Grouping: (3−4)+1(3−4)x(3x−4)+1(3x−4)

   Factoring out (3−4)(3x−4): (+1)(3−4)(x+1)(3x−4)

   The zeroes are =−1x=−1 and =43x=34​.

   Verification: The sum of the zeroes −1+43=13−1+34​=31​ matches −−13−3−1​, and the product −1×43=−43−1×34​=−34​ matches −433−4​.

   These detailed solutions through factorization confirm the relationships between the zeroes and the coefficients for each polynomial.

7. 7.Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.

   (i) 1/4, 1, (ii) root √2,1/3, (iii) 0, √5, (iv) 1,1, (v) -1/4, 1/4, (vi) 4,1

   
   

   Let's find a quadratic polynomial for each set of given sums and products of zeroes. We'll use the formula for a quadratic polynomial 2++ax2+bx+c, where the sum of the zeroes +=−α+β=−ab​ and the product of the zeroes =αβ=ac​.

   (i) Sum = 1441​, Product = 1

   Given:

   • +=14α+β=41​
   • =1αβ=1

   Choosing =1a=1 for simplicity:

   • =−(+)=−1×14=−14b=−a(α+β)=−1×41​=−41​
   • ==1×1=1c=a(αβ)=1×1=1

   The quadratic polynomial is: 2−14+1x2−41​x+1

   (ii) Sum = 22​, Product = 1331​

   Given:

   • +=2α+β=2​
   • =13αβ=31​

   Choosing =1a=1:

   • =−(+)=−1×2=−2b=−a(α+β)=−1×2​=−2​
   • ==1×13=13c=a(αβ)=1×31​=31​

   The quadratic polynomial is: 2−2+13x2−2​x+31​

   (iii) Sum = 0, Product = 55​

   Given:

   • +=0α+β=0
   • =5αβ=5​

   Choosing =1a=1:

   • =−(+)=−1×0=0b=−a(α+β)=−1×0=0
   • ==1×5=5c=a(αβ)=1×5​=5​

   The quadratic polynomial is: 2+5x2+5​

   (iv) Sum = 1, Product = 1

   Given:

   • +=1α+β=1
   • =1αβ=1

   Choosing =1a=1:

   • =−(+)=−1×1=−1b=−a(α+β)=−1×1=−1
   • ==1×1=1c=a(αβ)=1×1=1

   The quadratic polynomial is: 2−+1x2−x+1

   (v) Sum = -1/4, Product = 1/4

   Given:

   • +=−14α+β=−41​
   • =14αβ=41​

   Choosing =1a=1:

   • =−(+)=−1×−14=14b=−a(α+β)=−1×−41​=41​
   • ==1×14=14c=a(αβ)=1×41​=41​

   The quadratic polynomial is: 2+14+14x2+41​x+41​

   (vi) Sum = 4, Product = 1

   Given:

   • +=4α+β=4
   • =1αβ=1

   Choosing =1a=1:

   • =−(+)=−1×4=−4b=−a(α+β)=−1×4=−4
   • ==1×1=1c=a(αβ)=1×1=1

   The quadratic polynomial is: 2−4+1x2−4x+1

   In each case, the polynomial 2++ax2+bx+c is constructed to satisfy the given sum and product of its roots.

8. 8.Quick Revision

   1. Introduction to Polynomials: Polynomials are like mathematical sentences with numbers and letters, where the letters (like x or y) can stand for different numbers. For example, 2+3+2x2+3x+2 is a polynomial.

   2. Geometrical Meaning of the Zeroes of a Polynomial: Zeroes of a polynomial are the x-values where the polynomial's graph touches or crosses the x-axis (the flat line). These are the points where the value of the polynomial is zero.

   3. Graphical Method to Find Zeroes: You can draw the graph of a polynomial and see where it hits the x-axis. Those points where it hits are the zeroes. For example, if the graph touches the x-axis at =−1x=−1 and =2x=2, then -1 and 2 are the zeroes.

   4. Relationship Between Zeroes and Coefficients of a Polynomial: For a quadratic polynomial (which is a polynomial of degree 2, like 2++ax2+bx+c), there’s a cool connection:

   • The sum of its zeroes =−/=−b/a
   • The product of its zeroes =/=c/a

   5. Finding Zeroes of Quadratic Polynomials: To find the zeroes, you can:

   • Factor the polynomial, if possible.
   • Use the quadratic formula: =[−±2−4]/(2)x=[−b±b2−4ac​]/(2a)
   • And after finding the zeroes, you can add and multiply them to check the relationship with the coefficients.

   6. Finding a Quadratic Polynomial from Given Sum and Product of Zeroes: If you know the sum and product of the zeroes (let's say they are s and p), you can make a polynomial like this: 2−+x2−(sum of zeroes)x+(product of zeroes), which looks like 2−+x2−sx+p.