Question 183·200 Super-Hard SAT Math Questions·Advanced Math

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The polynomial $126x^4 + 245x^2 + 84$ can be written in the form $(k)(ax^2+b)(cx^2+d)$ , where $a$ , $b$ , $c$ , $d$ , and $k$ are integers.

Which choice gives the smallest possible value of $ab$ ?

A polynomial with only even powers, such as $x^4$ and $x^2$ , is a quadratic in $t=x^2$ . Graph that quadratic in Desmos to find its zeros, then use the zeros to build integer-coefficient factors. Keep any extra multiplier in $k$ when minimizing a product from one binomial.

Hints

Hint 1
Let $t=x^2$ . Then $x^4$ becomes $t^2$ , giving you a quadratic whose zeros you can find by graphing. What quadratic do you get?
Hint 2
An x-intercept is where a graph has height $0$ . Desmos may show a rounded decimal there, so test a likely fraction in the quadratic before using it to make a factor.
Hint 3
The zero of $at+b$ is $-b/a$ , so it fixes the ratio between $a$ and $b$ . To keep $ab$ small, where should an extra whole-number multiplier go?

Step-by-step

Graph the quadratic in $x^2$
Step 1
Turn the even-power expression into a quadratic
Because $x^4=(x^2)^2$ , let $t=x^2$ . The given expression becomes $126t^2+245t+84$ . This is a quadratic, a polynomial whose highest power of $t$ is $2$ . Its zeros will help you find the requested factors.
Step 2
Find the quadratic’s zeros
Type $g(x)=126x^2+245x+84$ in Desmos. The graph’s horizontal coordinate stands for our $t$ . Click its two x-intercepts, where the graph has height $0$ . Desmos shows $(-1.5,0)$ and approximately $(-0.444,0)$ .
Step 3
Check the exact fractions
The intercepts may be rounded. The one at $-1.5$ suggests $-\frac32$ ; the one near $-0.444$ suggests $-\frac49$ . Type $g(-\frac32)$ and $g(-\frac49)$ . Desmos prints $0$ for each, confirming both zeros exactly.
Step 4
Turn the zeros into factors
The zero of $at+b$ is $-b/a$ : that value makes the factor $0$ . So zeros $-\frac32$ and $-\frac49$ give the integer-coefficient factors $2t+3$ and $9t+4$ , respectively.
Step 5
Find the multiplier outside the factors
The leading term of $(2t+3)(9t+4)$ is $18t^2$ . Match the given leading coefficient: type $126/18$ , and Desmos prints $7$ . So $126t^2+245t+84=7(2t+3)(9t+4)$ .
Step 6
Put $x^2$ back into the factors
Replace $t$ with $x^2$ in both factors: $126x^4+245x^2+84=7(2x^2+3)(9x^2+4)$ . This has the required form, with the extra multiplier left in $k$ .
Step 7
Show why the smaller pair is the minimum
Could different integer factors give a smaller $ab$ ? The factor $ax^2+b$ corresponds to $at+b$ , whose zero is $-b/a$ . The quadratic has only the two zeros found above, so $(a,b)$ must be $(2m,3m)$ or $(9m,4m)$ for some nonzero integer $m$ . Each product grows with $m^2$ , so choose $|m|=1$ and the smaller pair, $(2,3)$ . The factorization above shows that pair is allowed.
Step 8
Calculate the requested product
For the smallest pair, $a=2$ and $b=3$ . Type $2\times3$ ; Desmos prints $6$ . So the smallest possible value of $ab$ is $6$ . Choice A.

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SAT Advanced AlgebraIntermediateCoreFactor algebraic expressions

Question 183·200 Super-Hard SAT Math Questions·Advanced Math

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