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A polynomial is a kind of algebraic expression and can be classified into distinct categories depending on their degree, such as linear polynomial (x), quadratic polynomial (x2), cubic polynomial (x3), and so on. The zeros of a polynomial can be determined by using the degree of the polynomial. The polynomial degree is equivalent to the number of zeros in the formula. A coefficient is a number or symbol in mathematics that represents a constant value multiplied by the variable of a single term or the terms of a polynomial. In the expression ax2 + bx + c, for example, x is the variable and a and b are the coefficients.
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Key terms: Relationship Between the Zeros and Coefficients, polynomial, algebraic expression, coefficient, linear polynomial, cubic polynomial, quadratic polynomial
Also read: Tangent to a Circle
Relationship Between the Zeros and Coefficients of a Polynomial
[Click Here for Sample Questions]
If P(a) = 0, a real integer "a" is a zero of a polynomial P(x). The Factor theorem helps to understand how to find a polynomial's zero. If a polynomial P(x) has a zero, then (x-k) is a factor of the polynomial. The following is the relationship between a zeros and its coefficients of polynomial:
Linear Polynomial
A linear polynomial is an expression in which the polynomial's degree is equal to 1. A linear polynomial in the form ax+b should be used. "x" is a variable here, while "a" and "b" are constants.
The zero of a polynomial is -b/a = – constant term/coefficient of x) if the polynomial P(x) is ax+b.
Quadratic Polynomial
A polynomial having the maximum degree of 2 is called a quadratic polynomial. A quadratic polynomial in the form ax2 + bx + c should be used. In this scenario, a ≠ 0 is appropriate. Assume α and β are the two zeros of a polynomial, then
The sum of zeros, α + β is -b/a = – Coefficient of x/ Coefficient of x2
The product of zeros, αβ is c/a = Constant term / Coefficient of x2
Cubic Polynomial
A polynomial having the maximum degree of 3 known as a cubic polynomial . The cubic polynomial must be written as ax3 + bx2 + cx + d, where a ≠ 0. Let’s say the three zeros of a polynomial are α, β, and γ, so
The sum of zeros, α + β + γ is -b/a = – Coefficient of x2/ coefficient of x3
The sum of the product of zeros, αβ+ βγ + αγ is c/a = Coefficient of x/Coefficient of x3
The product of zeros, αβγ is -d/a = – Constant term/Coefficient of x3
The video below explains this:
Polynomials Detailed Video Explanation:
Also read: Tangent to a Circle
Things to remember
- A polynomial is a kind of algebraic expression and they can also be classified into distinct categories depend on their degree, such as linear polynomial (x), quadratic polynomial (x2), cubic polynomial (x3), and so on.
- Anyone can easily determine the zeros of a polynomial using the degree of the polynomial.
- The polynomial degree is equivalent to the number of zeros in the formula.
- The Factor theorem helps to understand how to find a polynomial’s zero.
- If a polynomial P(x) has a zero, then (x-k) is a factor of the polynomial.
Also read: Isosceles Triangle Theorems
Sample Questions
Ques. Determine the sum and product of zeros of the quadratic polynomial 4x2 – 9. (4 marks)
Ans. Given equation is 4x2 – 9.
4x2 – 9 can be written as 2x2 – 33, which is equivalent to (2x+3)(2x-3).
To determine the zeros of a polynomial, first we have to equate the above expression to 0
(2x+3)(2x-3) = 0
2x+3 = 0
2x = -3
X = -3/2
Similarly, 2x-3 = 0,
2x = 3
x = 3/2
So, the zeros of a given quadratic polynomial is 3/2 and -3/2.
Determining the sum and product of a polynomial:
The sum of the zeros = (3/2)+ (-3/2) = (3/2)-(3/2) = 0
The product of zeros = (3/2).(-3/2) = -9/4.
Ques. Anjali has the idea that the zeros of a quadratic polynomial are -3 and 5. Now with this given information how can we determine the equation of the polynomial? (3 marks)
Ans. The zeros of the quadratic polynomial are -3 and 5.
Let α = -3, and β = 5
Then, we have the sum of the roots = α + β = 2
Product of the roots = α.β = -15
The required quadratic equation is x2 - (α + β) + α.β = 0
x2 - 2(x) + (-15) = 0
x2 - 2x - 15 = 0
Hence, the equation of the quadratic polynomial is x2 - 2x - 15 = 0
Ques. Given polynomial equation : 3x3 - 2x2 + 5x + 1. Now determine the sum of the squares of the zeroes of a polynomial? (3 marks)
Ans. The given polynomial expression is 3x3 - 2x2 + 5x + 1
The formulas for the zeros of the polynomials are as follows.
α + β + γ = - coefficient of x2/coefficient of x3 = -(-2)/3 = 2/3
αβ + βγ + γα = coefficient of x/coefficient of x3 = 5/3
(α + β + γ)2 = α2 + β2 + γ2 + 2(αβ + βγ + γα)
(2/3)2 = α2 + β2 + γ2 + 2(5/3)
4/9 = α2 + β2 + γ2 + 10/3
α2 + β2 + γ2 = 4/9 - 10/3
α2 + β2 + γ2 = 4/9 - 30/9
α2 + β2 + γ2 = -26/9
Therefore, the sum of the squares of the zeros of the polynomial is -26/9.
Ques. Find the value of x2–3x–4x2–3x–4 at x=−1x=−1 and x=4x=4? (3 marks)
Ans. For x=−1x=−1
p(–1)=(–1)2–3(–1)–4p(–1)=(–1)2–3(–1)–4
⇒1+3–4⇒1+3–4
⇒4–4⇒4–4
⇒0⇒0
Then, for x=4x=4
p(4)=(4)2–3(4)–4p(4)=(4)2–3(4)–4
⇒16–12–4⇒16–12–4
⇒16–16⇒16–16
⇒0⇒0
As p(–1)=0p(–1)=0 and p(4)=0,–1,p(4)=0,–1, and 44 are called the zeros of the quadratic polynomial x2–3x–4x2–3x–4. In general, a real number k is said to be a zero of a polynomial p(x)p(x), if pk=0pk=0.
Ques. Determine the zeros of the polynomial x2+7x+10x2+7x+10 (3 marks)
Ans. Given polynomial equation is x2+7x+10x2+7x+10
Using the process of splitting the middle terms, we may find the polynomial's zeros.
⇒x2+5x+2x+10=0⇒x2+5x+2x+10=0
⇒x(x+5)+2(x+5)=0⇒x(x+5)+2(x+5)=0
⇒(x+2)(x+5)=0⇒(x+2)(x+5)=0
⇒x=–2⇒x=–2 and x=–5x=–5
Hence, the zeroes of the given polynomial x2+7x+10x2+7x+10 are −2−2 and −5−5.
Ques. Find the zeroes of the polynomial p(x)=6x4–x3–15x2+2x–7p(x)=6x4–x3–15x2+2x–7. If x=2x=2. (4 marks)
Ans. The polynomial is p(x)=6x4–x3–15x2+2x–7
p(x)=6x4–x3–15x2+2x–7 and x=2x=2.
⇒6(2)4–23–15(2)2+2(2)–7
⇒6(2)4–23–15(2)2+2(2)–7
⇒6×16–8–15(4)+2(2)–7
⇒6×16–8–15(4)+2(2)–7
⇒6×16–8–15(4)+2(2)–7
⇒6×16–8–15(4)+2(2)–7
⇒96–8–60+4–7
⇒96–8–60+4–7
⇒100–75=25
⇒100–75=25
Therefore, the zeroes of the given polynomial is 25.
Ques. Find the zeros of the quadratic polynomial x2–2x–8x2–2x–8, and verify the relationship between the zeroes and the coefficient. (4 marks)
Ans. The given polynomial is x2–2x–8x2–2x–8.
Here, we can find the zeroes of the polynomial by the method of splitting the middle terms.
⇒x2–2x–8=0
⇒x2–2x–8=0
⇒x2–4x+2x–8=0
⇒x2–4x+2x–8=0
⇒x(x–4)+2(x–4)=0
⇒x(x–4)+2(x–4)=0
⇒(x+2)(x–4)=0
⇒(x+2)(x–4)=0
⇒x+2=0,x–4=0
⇒x+2=0,x–4=0
⇒x=–2,x=4
⇒x=–2,x=4
So, the zeroes of the given polynomial x2–2x–8x2–2x–8 are −2−2 and 44.
Sum of the zeroes =α+β=−ba=−( Coefficient of x) Coefficient of x2=α+β=−ba=−( Coefficient of x) Coefficient of x2
Product of the zeroes =α×β=ca= Constant term Coefficient of x2=α×β=ca= Constant term Coefficient of x2
⇒(–2)×4=−8⇒(–2)×4=−8
So, the sum and product of the zeroes are 22 and −8−8 respectively.
As a result, the relationship between the zeroes and the coefficients is verified.
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