Question

how are the binomial factors of $x^2 + 7x - 18$ and $x^2 - 7x - 18$ similar? how are they different? the trinomial $x^2 + 7x - 18$ factors into $\square$ and $x^2 - 7x - 18$ factors into $\square$. the signs in each binomial are $\blacktriangledown$ (factor completely.)

Explanation:

Step1: Factor \(x^2 + 7x - 18\)

We need two numbers that multiply to \(-18\) and add up to \(7\). The numbers are \(9\) and \(-2\) because \(9\times(-2)=-18\) and \(9 + (-2)=7\). So, \(x^2 + 7x - 18=(x + 9)(x - 2)\).

Step2: Factor \(x^2 - 7x - 18\)

We need two numbers that multiply to \(-18\) and add up to \(-7\). The numbers are \(-9\) and \(2\) because \((-9)\times2=-18\) and \(-9 + 2=-7\). So, \(x^2 - 7x - 18=(x - 9)(x + 2)\).

Step3: Analyze Similarities and Differences

• Similarities: Both factorizations use the same constant terms (\(9\) and \(2\)) in the binomials. The product of the constant terms in each binomial is \(-18\) (since \(9\times(-2)=-18\) and \((-9)\times2=-18\)).
• Differences: The signs of the linear terms (the terms with \(x\)) in the binomials are reversed. In \(x^2 + 7x - 18\), the binomials are \((x + 9)(x - 2)\) (positive \(9\) and negative \(2\)), while in \(x^2 - 7x - 18\), the binomials are \((x - 9)(x + 2)\) (negative \(9\) and positive \(2\)). Also, the middle term of the trinomial (the linear term) has opposite signs (\(+7x\) vs \(-7x\)) because the sum of the constant terms in the binomials is opposite (\(9+(-2)=7\) vs \(-9 + 2=-7\)).

Answer:

The trinomial \(x^2 + 7x - 18\) factors into \((x + 9)(x - 2)\) and \(x^2 - 7x - 18\) factors into \((x - 9)(x + 2)\). The signs in each binomial are reversed for the linear - term coefficients (or the signs of the constants in the binomials related to the linear term) because the middle term of the trinomial has opposite signs, leading to opposite sums of the constant terms in the binomial factors, while the product of the constant terms in the binomial factors remains the same (\(-18\)).