The expression above can be written as , where , , , and are integers. Which choice is the maximum possible value of ?

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Super-Hard SAT Math Question 57 | Free SAT Prep | Aniko

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Question 57·200 Super-Hard SAT Math Questions·Advanced Math

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20x^{12} + kx^6 + 45

The expression above can be written as $(ax^6 + b)(cx^6 + d)$ , where $a$ , $b$ , $c$ , and $d$ are integers. Which choice is the maximum possible value of $k$ ? Writt‍e⁠n‍ by Аnікo

When you see a polynomial like $20x^{12}+kx^6+45$ , treat $x^6$ as a single variable so the expression becomes a quadratic. For a factorization $(ay+b)(cy+d)$ , immediately match coefficients to get $ac$ and $bd$ , then write the middle coefficient as $k=ad+bc$ . A powerful next step is to use the fixed products to rewrite one cross-term in terms of the other (for example, express $ad$ using $bc$ ), turning the problem into maximizing a single expression. (Ani‌кo.aі)

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Use substitution to spot a quadratic pattern

Try letting $y=x^6$ so the expression looks like a quadratic in $y$ .

Match coefficients after multiplying

Expand $(ay+b)(cy+d)$ and match the $y^2$ term and the constant term to get two product equations.

Reduce the maximization to one variable

From $ac$ and $bd$ , see if you can rewrite $ad$ in terms of $bc$ (or vice versa), so $k=ad+bc$ depends on just one quantity. aniк⁠o‍.аi

Desmos Guide

Create a table for factor choices

Make a 4-column table with headers $b$ , $c$ , $a$ , $d$ . Enter integer values of $b$ that divide $45$ (such as 1, 3, 5, 9, 15, 45) and values of $c$ that divide $20$ (such as 1, 2, 4, 5, 10, 20).

Compute matching $a$ and $d$

In new columns, define $a=20/c$ and $d=45/b$ . Only keep rows where both $a$ and $d$ are integers.

Compute $k$ for each row and find the largest

Add a column for $k=a\cdot d+b\cdot c$ . Scan the $k$ values and identify the largest one. Аnі‍кo - Freе ЅАТ ‌Рrеp

Step-by-step Explanation

Rewrite as a quadratic in a single variable

Let $y=x^6$ . Then the expression becomes

20y^2+ky+45

If it factors as $(ay+b)(cy+d)$ , then expanding gives

(ay+b)(cy+d)=acy^2+(ad+bc)y+bd.

Match coefficients

Match coefficients with $20y^2+ky+45$ :

$ac=20$
$bd=45$
$k=ad+bc$

Express $k$ using a single product

To maximize $k$ , take $a,b,c,d$ with the same sign so that both $bc$ and $ad$ are positive.

Let $t=bc$ (a positive integer). From $ac=20$ and $bd=45$ ,

a=\frac{20}{c},\quad d=\frac{45}{b}.

ad=\frac{20}{c}\cdot\frac{45}{b}=\frac{900}{bc}=\frac{900}{t}.

Therefore,

k=ad+bc=\frac{900}{t}+t.

Maximize $t+\frac{900}{t}$ over possible integer values

Because $b$ and $c$ are integers with $bd=45$ and $ac=20$ , the value $t=bc$ must be a positive divisor of $900$ . In particular, $1\le t\le 900$ . Frοm аnікo.aі

Compare any such $t$ to the endpoint value $901$ :

901-\left(t+\frac{900}{t}\right)=\frac{901t-t^2-900}{t} =\frac{-(t-1)(t-900)}{t}.

For $1\le t\le 900$ , we have $(t-1)\ge 0$ and $(t-900)\le 0$ , so $(t-1)(t-900)\le 0$ , making the whole fraction nonnegative. Thus,

t+\frac{900}{t}\le 901.

Show the maximum is achievable and conclude

The value $901$ is achievable, for example with $t=bc=900$ by choosing $b=45$ and $c=20$ . Then $a=1$ and $d=1$ , and

(x^6+45)(20x^6+1)=20x^{12}+901x^6+45.

So the maximum possible value of $k$ is 901.

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Step-by-step Explanation

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