The quartic polynomial can be written in the form where and are real constants. What is the value of ?

Solution available with step-by-step explanation

SAT Equivalent Expressions Question 222 | Free SAT Prep | Aniko

00:00

Question 222·Hard·Equivalent Expressions

0 / 233

The quartic polynomial

x^4 - 5x^3 + kx^2 + 45x - 54

can be written in the form

(x^2 - ax + 6)(x^2 - bx - 9),

where $a$ and $b$ are real constants. What is the value of $k$ ?

Your answer

(Express the answer as an integer)

When a polynomial is given in both expanded and factored forms with unknown constants, expand the factored form symbolically first, then match coefficients of corresponding powers of $x$ . Use the simplest matching coefficients (often $x^3$ and $x$ ) to form a small linear system for the unknowns, solve it quickly with substitution or elimination, and finally plug those values into the expression for the target coefficient (here, the $x^2$ term) instead of expanding everything multiple times.

Hints

Click me!

Start by expanding the product

Do not plug in any numbers for $a$ and $b$ yet. First, multiply out $(x^2 - ax + 6)(x^2 - bx - 9)$ and collect like terms in powers of $x$ .

Use coefficient matching

Once you have the expanded form, line it up with $x^4 - 5x^3 + kx^2 + 45x - 54$ . Match the coefficients of $x^3$ and $x$ to create equations involving $a$ and $b$ , and match the $x^2$ coefficient to express $k$ in terms of $a$ and $b$ .

Solve the system for a and b

From the $x^3$ and $x$ coefficients you should get two linear equations in $a$ and $b$ . Use substitution or elimination to find $a$ and $b$ , then plug them into your expression for the $x^2$ coefficient to get $k$ .

Desmos Guide

Let Desmos expand the factored form

In one expression line, type (x^2 - a x + 6)(x^2 - b x - 9). Desmos will automatically expand this and show the coefficients in terms of a and b (including the coefficient of x^2).

Use Desmos to solve for a and b

In two new lines, type a + b = 5 and 9a - 6b = 45. Desmos will plot these as lines in the (a,b)-plane; look for their intersection point to read the values of a and b.

Read off the value of k

Go back to the expanded product from step 1 and substitute the intersection values of a and b (or just update the sliders). The coefficient of x^2 in that expanded expression is exactly the value of k; read that number from Desmos.

Step-by-step Explanation

Expand the factored form

Write and expand the product:

(x^2 - ax + 6)(x^2 - bx - 9) = x^2(x^2 - bx - 9) - ax(x^2 - bx - 9) + 6(x^2 - bx - 9) \\[0.7em] = x^4 - bx^3 - 9x^2 - ax^3 + abx^2 + 9ax + 6x^2 - 6bx - 54 \\[0.7em] = x^4 -(a+b)x^3 + (ab - 3)x^2 + (9a - 6b)x - 54.

So the polynomial in factored form is equivalent to

x^4 -(a+b)x^3 + (ab - 3)x^2 + (9a - 6b)x - 54.

Match coefficients with the given quartic

We are told this equals

x^4 - 5x^3 + kx^2 + 45x - 54.

Because the polynomials are equal for all $x$ , the coefficients of each power of $x$ must match:

$x^3$ term: $-(a+b) = -5$ , so $a + b = 5$ .
$x$ term: $9a - 6b = 45$ .
$x^2$ term: $ab - 3 = k$ (this is the expression we will use to find $k$ ).

The constant term $-54$ already matches automatically.

Solve for a and b using the easier equations

Use the two equations involving $a$ and $b$ :

$a + b = 5$
$9a - 6b = 45$

From $a + b = 5$ , write $b = 5 - a$ and substitute into the second equation:

9a - 6(5 - a) = 45 \\[0.7em] 9a - 30 + 6a = 45 \\[0.7em] 15a = 75 \\[0.7em] a = 5.

Then

b = 5 - a = 5 - 5 = 0.

Find k from the x² coefficient

Recall the relationship from the $x^2$ term:

k = ab - 3.

Substitute $a = 5$ and $b = 0$ :

k = 5 \cdot 0 - 3 = -3.

So the value of $k$ is $-3$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation