Problem

Use zeros \(-3\) and \(-4\) to write the monic quadratic in factored form. Expand it, compare the \(x\)-coefficient with \(x^{2}+bx+12\), find \(b\), and verify all terms of the polynomial.

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What this problem is really about

The given zeros determine a monic quadratic’s two factors. We’ll convert each negative zero into its x minus r factor, expand the product, match the linear coefficient with the unknown parameter, and verify the result using both the root-sum and root-product relationships.

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Using only root −3 gives b = 3. Both zeros are needed to determine the polynomial.

The root sum is −7, so b = −7. The coefficient relation is −b = −7, which gives b = 7.

The root product is 12, so b = 12. The product determines the constant term, not the x-coefficient.

The root sum is −7, so −b = −7 and b = 7. The factors (x + 3)(x + 4) expand to x² + 7x + 12.

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Use zeros \(-3\) and \(-4\) to write the monic quadratic in factored form. Expand it, compare the \(x\)-coefficient with \(x^{2}+bx+12\), find \(b\), and verify all terms of the polynomial.

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