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Factor the following expression. \[ \begin{array}{c}2 x^{2}+19 x+17 \\ (x+[?])(\square x+\square)\end{array} \]

Ask by Macdonald Dickson. in the United States
Jan 21,2025

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Tutor-Verified Answer

Answer

The factored form is \((x+1)(2x+17)\).

Solution

Factor the expression \( 2x^2+19x+17 \). Factor the expression by following steps: - step0: Factor: \(2x^{2}+19x+17\) - step1: Rewrite the expression: \(2x^{2}+\left(17+2\right)x+17\) - step2: Rewrite the expression: \(2x^{2}+17x+2x+17\) - step3: Factor the expression: \(x\left(2x+17\right)+2x+17\) - step4: Factor the expression: \(\left(x+1\right)\left(2x+17\right)\) The factored form of the expression \(2x^2+19x+17\) is \((x+1)(2x+17)\).

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Bonus Knowledge

To factor the expression \(2x^{2} + 19x + 17\), we first look for two numbers that multiply to \(2 \cdot 17 = 34\) and add up to \(19\). The numbers \(17\) and \(2\) fit this requirement since \(17 \cdot 2 = 34\) and \(17 + 2 = 19\). Now we can rewrite the middle term \(19x\) using these two numbers: \[ 2x^2 + 17x + 2x + 17 \] Next, we can factor by grouping: \[ = (2x^2 + 17x) + (2x + 17) \] From the first group \(2x^2 + 17x\), we can factor out \(x\): \[ = x(2x + 17) \] From the second group \(2x + 17\), we can factor out \(1\): \[ = 1(2x + 17) \] Now we combine both parts: \[ = (2x + 17)(x + 1) \] Thus, the factored form of the expression is: \[ (2x + 17)(x + 1) \]

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