QuestionAugust 25, 2025

A polynomial with leading term x^3 has 5 and 7 as roots; 7 is a double root. What is this polynomial in standard form?

A polynomial with leading term x^3 has 5 and 7 as roots; 7 is a double root. What is this polynomial in standard form?
A polynomial with leading term x^3 has 5 and 7 as roots; 7 is a double root. What is this polynomial in standard form?

Solution
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Answer

x^3 - 19x^2 + 119x - 245 Explanation 1. Identify the roots The polynomial has roots at x = 5 and x = 7, with x = 7 being a double root. 2. Construct the factors The factors corresponding to these roots are (x - 5) and (x - 7)^2. 3. Multiply the factors Multiply the factors to form the polynomial: (x - 5)(x - 7)^2. 4. Expand the expression First, expand (x - 7)^2: (x - 7)^2 = x^2 - 14x + 49. Then multiply by (x - 5): (x - 5)(x^2 - 14x + 49) = x(x^2 - 14x + 49) - 5(x^2 - 14x + 49). 5. Simplify the expression Expand each term: x(x^2 - 14x + 49) = x^3 - 14x^2 + 49x. -5(x^2 - 14x + 49) = -5x^2 + 70x - 245. Combine like terms: x^3 - 19x^2 + 119x - 245.

Explanation

1. Identify the roots<br /> The polynomial has roots at $x = 5$ and $x = 7$, with $x = 7$ being a double root.<br /><br />2. Construct the factors<br /> The factors corresponding to these roots are $(x - 5)$ and $(x - 7)^2$.<br /><br />3. Multiply the factors<br /> Multiply the factors to form the polynomial: $(x - 5)(x - 7)^2$.<br /><br />4. Expand the expression<br /> First, expand $(x - 7)^2$: $(x - 7)^2 = x^2 - 14x + 49$.<br /> Then multiply by $(x - 5)$: <br /> $(x - 5)(x^2 - 14x + 49) = x(x^2 - 14x + 49) - 5(x^2 - 14x + 49)$.<br /><br />5. Simplify the expression<br /> Expand each term: <br /> $x(x^2 - 14x + 49) = x^3 - 14x^2 + 49x$.<br /> $-5(x^2 - 14x + 49) = -5x^2 + 70x - 245$.<br /> Combine like terms: $x^3 - 19x^2 + 119x - 245$.
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