Super-Hard SAT Advanced Math Question 2 | Free SAT Prep

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Question 2·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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A polynomial $P$ can be written as $P(x)=(x^2+mx+1)(x^2+nx+1)$ , where $m$ and $n$ are positive integers. The value of $P(1)$ is $360$ . When $P(x)$ is written in standard form, its $x^2$ coefficient is $b$ . What is the greatest possible value of $b$ ?

Your answer

For a polynomial given in factored form, avoid fully expanding unless necessary. First use the stated value of the polynomial to create a relationship between the factor parameters. Then express the requested coefficient in terms of those parameters. Here, shifting the variables by $2$ turns the condition into an integer factor-pair problem, and the factor pair closest together produces the smallest sum and therefore the greatest coefficient.

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Evaluate the factored expression

Substitute $x=1$ into both factors. This creates a product of two positive integers that equals $360$ .

Track the needed coefficient

When the two quadratics are multiplied, the $x^2$ coefficient comes from the product of the two middle coefficients and the two constant-leading products.

Compare factor pairs

Rewrite the coefficient using $r=m+2$ and $s=n+2$ . Then determine which factor pair of $360$ makes $r+s$ smallest.

Desmos Guide

Use algebra first

Algebra is faster because the key step is recognizing that $(m+2)(n+2)=360$ . Desmos can verify the closest integer factor pair.

Locate nearby factor values

Graph $y=360/x$ and $y=x$ to see that equal factors would be near $19$ . In a table, enter integer $x_1$ values from $15$ through $20$ and enter $y_1=360/x_1$ . Identify the closest pair of whole-number values whose product is $360$ .

Verify the coefficient

Enter $(18-2)(20-2)+2$ to verify the resulting $x^2$ coefficient.

Step-by-step Explanation

Use the value of the polynomial at 1

Substitute $x=1$ into the factored form: $P(1)=(1+m+1)(1+n+1)=(m+2)(n+2)$ . Since $P(1)=360$ , $(m+2)(n+2)=360$ .

Express the coefficient in terms of the factors

Expanding only enough to find the $x^2$ coefficient gives $b=mn+2$ . Let $r=m+2$ and $s=n+2$ . Then $rs=360$ , and $b=(r-2)(s-2)+2$ . Because $rs=360$ , this becomes $b=366-2(r+s)$ .

Minimize the sum of the factor pair

To make $b=366-2(r+s)$ as large as possible, make $r+s$ as small as possible. Among positive integer factor pairs of $360$ , the pair closest together is $18$ and $20$ . Thus $m=16$ and $n=18$ , so $b=(16)(18)+2=290$ .

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Step-by-step Explanation

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Step-by-step Explanation