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2026-01-01
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2026-02-28
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<p>In this section, we will discuss the<a>relation</a>between the coefficients and the sum and product of the zeros of the polynomial. Let the roots of a quadratic polynomial in the form f(x) = ax2 + bx + c be α and β. If the roots are α and β, then the sum is: α + β = -b/a. Their product is: αβ = c/a.</p>
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<p>In this section, we will discuss the<a>relation</a>between the coefficients and the sum and product of the zeros of the polynomial. Let the roots of a quadratic polynomial in the form f(x) = ax2 + bx + c be α and β. If the roots are α and β, then the sum is: α + β = -b/a. Their product is: αβ = c/a.</p>
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<p>For example, finding the sum and product of zeros in f(x) = 2x2 + 5x + 3 Here, a = 2, b = 5, and c = 3.</p>
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<p>For example, finding the sum and product of zeros in f(x) = 2x2 + 5x + 3 Here, a = 2, b = 5, and c = 3.</p>
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<p>Using the formula to find the sum and product of the roots: α + β = -b/a and αβ = c/a α + β = -5/2 αβ = 3/2</p>
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<p>Using the formula to find the sum and product of the roots: α + β = -b/a and αβ = c/a α + β = -5/2 αβ = 3/2</p>
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<p>Let’s verify the answer by finding the roots of the quadratic polynomial. To find the roots, we use the quadratic formula: x = -b ± b2 - 4ac2a x = -5 ± 52 - 4 × 2 × 32 × 2 x = -5 ± 25 - (4 × 2 × 3)4 x = -5 ± 25 - 244 x = -5 ± 14 x = -5 + 14 or x = -5 - 14 x = -4/4 = -1 or x = -6/4 = -3/2 So, x = -1 and x = -3/2 The sum of the roots = -1 + (-3/2) = -5/2 The product of the roots = -1 × (-3/2) = 3/2 Therefore, the sum and product of the roots are directly related to the coefficients. </p>
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<p>Let’s verify the answer by finding the roots of the quadratic polynomial. To find the roots, we use the quadratic formula: x = -b ± b2 - 4ac2a x = -5 ± 52 - 4 × 2 × 32 × 2 x = -5 ± 25 - (4 × 2 × 3)4 x = -5 ± 25 - 244 x = -5 ± 14 x = -5 + 14 or x = -5 - 14 x = -4/4 = -1 or x = -6/4 = -3/2 So, x = -1 and x = -3/2 The sum of the roots = -1 + (-3/2) = -5/2 The product of the roots = -1 × (-3/2) = 3/2 Therefore, the sum and product of the roots are directly related to the coefficients. </p>
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<p>Let’s understand how the sum and product of a quadratic polynomial’s roots are related to its coefficients. The<a>factored form</a>of a quadratic polynomial is: f(x) = (x - a)(x - b), where a and b are the roots. Let the sum of the roots be S = a + b and the product be P = ab. Expanding the equation: (x - a)(x - b) = x2 - ax - bx + ab = x2 - (a + b)x + ab So, P(x) = x2 - Sx + P. Here, the<a>coefficient</a>of x is -S, so it is the negative of the sum of the roots and the term P is the product of the roots. </p>
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<p>Let’s understand how the sum and product of a quadratic polynomial’s roots are related to its coefficients. The<a>factored form</a>of a quadratic polynomial is: f(x) = (x - a)(x - b), where a and b are the roots. Let the sum of the roots be S = a + b and the product be P = ab. Expanding the equation: (x - a)(x - b) = x2 - ax - bx + ab = x2 - (a + b)x + ab So, P(x) = x2 - Sx + P. Here, the<a>coefficient</a>of x is -S, so it is the negative of the sum of the roots and the term P is the product of the roots. </p>
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<p>Now, let’s generalize the result for any quadratic polynomial of the form: f(x) = ax2 + bx + c. where ‘a’ is the coefficient of x2, b is the coefficient of x, and c is the constant. Consider the root as m and n, then the polynomial can be factorized as: f(x) = a(x - m)(x - n)</p>
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<p>Now, let’s generalize the result for any quadratic polynomial of the form: f(x) = ax2 + bx + c. where ‘a’ is the coefficient of x2, b is the coefficient of x, and c is the constant. Consider the root as m and n, then the polynomial can be factorized as: f(x) = a(x - m)(x - n)</p>
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<p>Let’s compare two forms of a quadratic polynomial: ax2 + bx + c and a(x - m)(x - n) Now divide both sides by ‘a’ to simplify: x2 + (b/a)x + c/a = (x - m)(x - n) Next, expand the right-hand side: (x - m)(x - n) = x2 - (m + n)x + mn So now we have: x2 + (b/a)x + c/a = x2 - (m + n)x + mn Since both expressions represent the same polynomial, we can<a>match</a>the coefficients: The coefficient of x: b/a = -(m + n) m + n = -b/a The constant term: c/a = mn So we’ve shown: Sum of the roots: m + n = -b/a </p>
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<p>Let’s compare two forms of a quadratic polynomial: ax2 + bx + c and a(x - m)(x - n) Now divide both sides by ‘a’ to simplify: x2 + (b/a)x + c/a = (x - m)(x - n) Next, expand the right-hand side: (x - m)(x - n) = x2 - (m + n)x + mn So now we have: x2 + (b/a)x + c/a = x2 - (m + n)x + mn Since both expressions represent the same polynomial, we can<a>match</a>the coefficients: The coefficient of x: b/a = -(m + n) m + n = -b/a The constant term: c/a = mn So we’ve shown: Sum of the roots: m + n = -b/a </p>
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