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The polynomial
p(x)=x^(3)+3x^(2)-4 has a known factor of
(x-1).
Rewrite
p(x) as a product of linear factors.

p(x)=

The polynomial $p(x)=x^{3}+3 x^{2}-4$ has a known factor of $(x-1)$ . $\newline$ Rewrite $p(x)$ as a product of linear factors. $\newline$ $p(x)=$

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Find the inverse of a linear function

Full solution

Q. The polynomial

p(x)=x^{3}+3 x^{2}-4

has a known factor of

(x-1)

.

\newline

Rewrite

p(x)

as a product of linear factors.

\newline

p(x)=

Factor Finding: Since we know that $(x - 1)$ is a factor of $p(x)$ , we can use polynomial long division or synthetic division to divide $p(x)$ by $(x - 1)$ to find the other factors.
Synthetic Division: Let's perform synthetic division using the root corresponding to the factor $(x - 1)$ , which is $x = 1$ . $\newline$ We set up the synthetic division as follows: $\newline$ \begin{array}{r|rrrr} 1 & 1 & 3 & 0 & -4 \ & & 1 & 4 & 4 \ \hline & 1 & 4 & 4 & 0 \end{array} $\newline$ The numbers on the bottom row, after the line, give us the coefficients of the quotient polynomial.
Quotient Polynomial: The quotient polynomial is $x^2 + 4x + 4$ . Since the remainder is $0$ , this confirms that $(x - 1)$ is indeed a factor of $p(x)$ .
Factoring Quadratic Polynomial: Now we need to factor the quadratic polynomial $x^2 + 4x + 4$ . This is a perfect square trinomial, which can be factored as $(x + 2)^2$ .
Final Expression: Therefore, we can express $p(x)$ as a product of its linear factors: $p(x) = (x - 1)(x + 2)^2$ .

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