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How are the binomial factors The trinomial \( x^{2}+7 x-18 \) fac (Factor completely)

Ask by Grant Huff. in the United States
Mar 11,2025

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\( x^{2} + 7x - 18 = (x - 2)(x + 9) \)

Solución

Factor the expression by following steps: - step0: Factor: \(x^{2}+7x-18\) - step1: Rewrite the expression: \(x^{2}+\left(9-2\right)x-18\) - step2: Calculate: \(x^{2}+9x-2x-18\) - step3: Rewrite the expression: \(x\times x+x\times 9-2x-2\times 9\) - step4: Factor the expression: \(x\left(x+9\right)-2\left(x+9\right)\) - step5: Factor the expression: \(\left(x-2\right)\left(x+9\right)\) The trinomial \( x^{2} + 7x - 18 \) can be factored completely into the binomial factors: \[ (x - 2)(x + 9) \]

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To factor the trinomial \( x^{2}+7x-18 \), we need to find two numbers that multiply to \(-18\) (the constant term) and add up to \(7\) (the coefficient of \(x\)). Those numbers are \(9\) and \(-2\). Therefore, we can rewrite the trinomial as: \[ x^{2} + 9x - 2x - 18 \] Now, we can group the terms: \[ (x^{2} + 9x) + (-2x - 18) \] Factoring each group gives us: \[ x(x + 9) - 2(x + 9) \] Now, factor out the common binomial factor \((x + 9)\): \[ (x + 9)(x - 2) \] Thus, the complete factorization of the trinomial \( x^{2}+7x-18 \) is \((x + 9)(x - 2)\). For a fun fact, did you know that the trinomial could be visualized as a mini puzzle where you need to find the right pieces (numbers) to fit perfectly together? Just like a puzzle, every trinomial has its unique pair of factors waiting to be discovered! When dealing with factorizations like this, a common mistake is to mix signs; make sure to keep track of positive and negative numbers carefully. Double-check your multiplication and addition outcomes – it’s easy to overlook one little detail that can lead you astray!

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