Rewrite the expression as a product of four linear factors: (3x^(2)-5x)^(2)-20(3x^(2)-5x)+96 Answer:

Question

Rewrite the expression as a product of four linear factors:  (3x^(2)-5x)^(2)-20(3x^(2)-5x)+96 Answer:

Bytelearn · Accepted Answer

Identify Given Expression: Identify the given expression and recognize that it resembles a quadratic in form, where the quadratic term is $(3x^2 - 5x)^2$, the linear term is $-20(3x^2 - 5x)$, and the constant term is $+96$. We can treat $3x^2 - 5x$ as a single variable, say $u$, and rewrite the expression as a quadratic in terms of $u$. Let $u = 3x^2 - 5x$. Then the expression becomes $u^2 - 20u + 96$.Factor Quadratic Expression: Factor the quadratic expression $u^2 - 20u + 96$. We are looking for two numbers that multiply to $96$ and add up to $-20$. These numbers are $-12$ and $-8$, since $-12 \times -8 = 96$ and $-12 + -8 = -20$. So, $u^2 - 20u + 96$ factors to $(u - 12)(u - 8)$.Substitute Back and Expand: Substitute back $3x^2 - 5x$ for $u$ in the factored form. We get $(3x^2 - 5x - 12)(3x^2 - 5x - 8)$.Factor First Quadratic: Now, we need to factor each quadratic expression $(3x^2 - 5x - 12)$ and $(3x^2 - 5x - 8)$ into two linear factors. Starting with $(3x^2 - 5x - 12)$, we look for two numbers that multiply to $3 \times -12 = -36$ and add up to $-5$. These numbers are $-9$ and $+4$, since $-9 \times 4 = -36$ and $-9 + 4 = -5$. So, $3x^2 - 5x - 12$ factors to $(3x + 4)(x - 3)$.Factor Second Quadratic: Next, factor $(3x^2 - 5x - 8)$. We look for two numbers that multiply to $3 \times -8 = -24$ and add up to $-5$. These numbers are $-6$ and $+1$, since $-6 \times 1 = -24$ and $-6 + 1 = -5$. So, $3x^2 - 5x - 8$ factors to $(3x + 1)(x - 8)$.Combine Linear Factors: Combine the linear factors from the previous steps to express the original expression as a product of four linear factors. The final factored form is $(3x + 4)(x - 3)(3x + 1)(x - 8)$.

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}-5 x\right)^{2}-20\left(3 x^{2}-5 x\right)+96$ $\newline$ Answer:

Full solution

More problems from Operations with rational exponents

Related topics

Rewrite the expression as a product of four linear factors:\newline(3x2−5x)2−20(3x2−5x)+96 \left(3 x^{2}-5 x\right)^{2}-20\left(3 x^{2}-5 x\right)+96 (3x2−5x)2−20(3x2−5x)+96\newlineAnswer:

Rewrite the expression as a product of four linear factors: $\newline$ $\left(3 x^{2}-5 x\right)^{2}-20\left(3 x^{2}-5 x\right)+96$ $\newline$ Answer: