Question

select the correct answer from each drop-down menu. simplify the rational expression \\(\frac{x^2 + 9x + 20}{x + 7}\\). the quotient is \\(\square\\), and the remainder is \\(\square\\).

Explanation:

Step1: Factor the numerator

We factor the quadratic expression in the numerator \(x^{2}+9x + 20\). We need to find two numbers that multiply to \(20\) and add up to \(9\). The numbers are \(4\) and \(5\), so \(x^{2}+9x + 20=(x + 4)(x+5)\). But we can also use polynomial long division. Let's divide \(x^{2}+9x + 20\) by \(x + 7\).

Using the formula for dividing a quadratic \(ax^{2}+bx + c\) by a linear term \(x + d\). We know that \(x^{2}+9x + 20=(x + 7)(x + 2)+6\). Let's verify: \((x + 7)(x + 2)=x^{2}+2x+7x + 14=x^{2}+9x+14\), then \(x^{2}+9x + 20-(x^{2}+9x + 14)=6\).

Step2: Determine quotient and remainder

When we divide a polynomial \(f(x)\) by \(g(x)=x + a\), we can write \(f(x)=g(x)q(x)+r\), where \(q(x)\) is the quotient and \(r\) is the remainder (a constant since the degree of \(g(x)\) is 1).

From the division \(x^{2}+9x + 20=(x + 7)(x + 2)+6\), the quotient \(q(x)=x + 2\) and the remainder \(r = 6\).

Answer:

The quotient is \(x + 2\), and the remainder is \(6\)