The polynomial p(x)=5x^(3)-44x^(2)+61 x+14 has a known factor of (x-7). Rewrite p(x) as a product of linear factors. p(x)=

Perform Polynomial Division: Step 1: Perform polynomial division to divide p(x) by (x-7). Calculation: Using synthetic division with root 7, 5x^3 - 44x^2 + 61x + 14 divided by x - 7: 7 | 5 -44 61 14 | 35 -63 -14 ----------------- 5 -9 -2 0 Result: 5x^2 - 9x - 2Factorize Quadratic Equation: Step 2: Factorize the quadratic equation 5x^2 - 9x - 2. Calculation: Using the AC method, where A=5, B=-9, C=-2, Product AC = -10, Sum B = -9, Factors of -10 that add up to -9 are -10 and 1, Rewrite middle term: 5x^2 - 10x + x - 2, Group: (5x^2 - 10x) + (x - 2), Factor out common terms: 5x(x - 2) + 1(x - 2), Factor by grouping: (5x + 1)(x - 2)Write as Linear Factors: Step 3: Write p(x) as a product of linear factors. Calculation: p(x) = (x - 7)(5x + 1)(x - 2)

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The polynomial $\newline$ $p(x)=5x^{3}-44x^{2}+61x+14$ has a known factor of $\newline$ $(x-7)$ . $\newline$ Rewrite $\newline$ $p(x)$ as a product of linear factors. $\newline$ $p(x)=$

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Q. The polynomial

\newline

p(x)=5x^{3}-44x^{2}+61x+14

has a known factor of

\newline

(x-7)

\newline

Rewrite

\newline

p(x)

as a product of linear factors.

\newline

p(x)=

Perform Polynomial Division: Step $1$ : Perform polynomial division to divide $p(x)$ by $(x-7)$ . $\newline$ Calculation: Using synthetic division with root $7$ , $\newline$ $5x^3 - 44x^2 + 61x + 14$ divided by $x - 7$ : $\newline$ \begin{array}{r|rrrr} 7 & 5 & -44 & 61 & 14 \ \hline & & 35 & -63 & -14 \ \hline & 5 & -9 & -2 & 0 \end{array} $\newline$ Result: $5x^2 - 9x - 2$
Factorize Quadratic Equation: Step $2$ : Factorize the quadratic equation $5x^2 - 9x - 2$ . $\newline$ Calculation: Using the AC method, where $A=5$ , $B=-9$ , $C=-2$ , $\newline$ Product $AC = -10$ , Sum $B = -9$ , $\newline$ Factors of $-10$ that add up to $-9$ are $-10$ and $1$ , $\newline$ Rewrite middle term: $A=5$ $0$ , $\newline$ Group: $A=5$ $1$ , $\newline$ Factor out common terms: $A=5$ $2$ , $\newline$ Factor by grouping: $A=5$ $3$
Write as Linear Factors: Step $3$ : Write $p(x)$ as a product of linear factors. $\newline$ Calculation: $p(x) = (x - 7)(5x + 1)(x - 2)$