Solve higher-degree equations from structure

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Lesson progressPractice problems 0/3

Difficulty: Advanced
Estimated time: 29 minutes
Domains: Advanced Math
Techniques: Higher-degree-equationsPolynomial-structureZero-product-propertySubstitutionComplete-root-recovery

What you’ll learn

Spot the structure that turns a big polynomial equation into smaller ones.
Graph a short all-number equation in Desmos without expanding it, and read every $x$ -intercept before you pick the one the question wants.
Break a product or a repeated expression into equations you already know how to solve.
Keep a solution of $0$ instead of dividing it away.
Bring every value back to $x$ after a substitution.
Collect every distinct real root before you answer what the question actually asks.

Prerequisites

Why this matters on the SAT

Turn one higher-degree equation into smaller equations

On the SAT, the structure you need is often in plain sight. Either a product equals $0$ , or the same expression shows up more than once. Here's the first kind.

Solution to the example

The left side is a product, and it equals $0$ . A product can only be $0$ when one of its factors is $0$ , so

x^2-25=0 \quad\text{or}\quad 2x+3=0.

The first factor is a difference of squares:

x^2-25=(x-5)(x+5),

so it gives $x=5$ or $x=-5$ . The second factor gives

x=-\frac32.

Only $-\frac32$ is among the choices, so the answer is B. Multiplied out, this equation has an $x^3$ term. But it came apart into one quadratic and one linear equation, and you already know how to solve both.

SAT example

Which choice is a solution to the equation

(x^2-25)(2x+3)=0?

A
$-25$
B
$-\frac32$
C
$\frac32$
D
$25$

Example: Find every real root in Desmos

When an equation has only numbers in it and asks for a root, a graph can often find every solution at once.

Worked example

Which choice is the greatest real solution to

(x^2-7x)^2+16(x^2-7x)+60=0?

A
$2$
B
$5$
C
$6$
D
$10$

Step 1

Graph the polynomial

The equation already equals $0$ . So its real solutions are the $x$ -values that make the left side $0$ , and on a graph those are the $x$ -intercepts. Type the left side into Desmos as it is, without expanding:

y=(x^2-7x)^2+16(x^2-7x)+60

Leaving $x^2-7x$ in its parentheses saves typing, and it spares you from multiplying out a fourth-degree polynomial.

Calculator loads as you approach

Click the curve to show its gray points, then click every x-intercept.

Step 2

Read every x-intercept

Click the curve, or its line in the expression list, and gray points of interest appear. Click each one that sits on the $x$ -axis. Their $x$ -coordinates are

x=1,\ 2,\ 5,\ 6.

Write down all four before you go back to the question.

Step 3

Pick the greatest

The question wants the greatest real solution. Out of $1$ , $2$ , $5$ , and $6$ , that's

\boxed{6}.

The answer is C.

Step 4

Know when the graph is enough

Why could you trust the graph here? The intercepts landed on whole numbers, and every answer choice is a different whole number, so there was nothing to guess. If the question had asked for a constant like $k$ , an exact answer with letters in it, or a boundary the graph can only approximate, you'd use structural algebra instead. You'll see how below.

Check your understanding:

Why read all four x-intercepts before choosing the greatest real solution?

Common mistake:

Answering from the first intercept you notice. Three of the four choices here are real roots, so a partial read can land on $2$ or $5$ , and either one checks out in the equation. List every $x$ -coordinate first, and only then apply the word that matters, like greatest, least, or positive.

Choose graphing or structure

Most of these equations give you something to work with: a product equal to $0$ , a factor of $x$ in every term, an expression that repeats, or only even powers like $x^4$ and $x^2$ . What you do with that structure depends on what the question wants you to find:

When a short all-number equation asks for a root, like the least, greatest, positive, negative, or a possible one, get $0$ on one side, graph the other side without expanding, and read every $x$ -intercept. The same works for a multiple-choice question when you can pick out the choices on the graph.
When the factors are already in front of you, like in $x(x-3)(x+4)=0$ , solve each one by hand if that's quicker than graphing.
When the question asks for a constant like $k$ , an answer with letters in it, an exact boundary, or proof that you've found every root, use structural algebra so every branch and boundary comes out exact.

A graph has limits, though. Its labels are numbers, not exact expressions. Roots that sit close together or far off to one side are easy to miss, so move the window until you can see everywhere the curve meets the $x$ -axis. And a rounded label like $1.414$ can point you to $\sqrt2$ , but it can't show that the value is exactly $\sqrt2$ , or pin down an exact boundary for a constant. Once you've read every intercept, match those numbers to the answer choice or to the value the question asks for.

Solve it with structural algebra

Structural algebra breaks one big equation into smaller ones. We'll call each smaller equation a branch. If

A(x)B(x)=0,

the branches are

A(x)=0 \quad\text{or}\quad B(x)=0.

A branch can give more than one value of $x$ , the way $x^2-25=0$ gave both $5$ and $-5$ in the first example. The whole method fits in one line:

\boxed{\text{reduce the structure}\ \longrightarrow\ \text{solve every branch}\ \longrightarrow\ \text{restore every }x.}

Step by step, it looks like this:

Get $0$ on one side, if it isn't there already.
Find the structure. Factor, or give the whole repeated expression a short name, like $u$ .
Solve every branch, one for each factor or each value of $u$ .
Go back to $x$ . Put the full expression back in for $u$ in every branch, and solve.
Combine the results. Drop repeated roots, keep every different real root, and only then apply words like least, positive, or sum.
Make sure nothing's missing. Check that you solved every factor and every value of $u$ . Plugging a root back in is a good check when it's quick.

Keep the zero-factor cases

Take

x(x^2-7x+10)=0.

It's tempting to divide both sides by $x$ to make it shorter. That leaves $x^2-7x+10=0$ , which gives $2$ and $5$ , and the root $0$ is gone. Dividing by $x$ quietly assumes $x\ne0$ , and that's exactly the case you still needed to check.

Let the zero-product property split it instead:

x=0 \quad\text{or}\quad x^2-7x+10=0.

Factor the quadratic branch:

x^2-7x+10=(x-2)(x-5).

So the complete solution set is

\boxed{x=0,\ 2,\ 5}.

Common mistake:

Dividing both sides by any expression with $x$ in it, not only $x$ itself. Dividing by $x-3$ or $x^2+x$ can throw away the solutions that make that expression $0$ , and the shorter work hides the loss. Factor first and set every factor with $x$ in it equal to $0$ . Divide only inside a branch where you know the divisor can’t be $0$ .

The same trap shows up after a substitution. If you reach

u(u-6)=0,

keep both

u=0 \quad\text{and}\quad u=6.

Don't cancel the $u$ . The branch $u=0$ can still give you one or more values of $x$ .

Use substitution for a repeated expression

Suppose the equation is

(x^2+x)^2-8(x^2+x)+12=0.

The whole expression $x^2+x$ shows up twice: once squared, and once multiplied by $-8$ . So treat it as one quantity and give it a short name. Let

u=x^2+x.

Now the equation is an ordinary quadratic,

u^2-8u+12=0,

which factors as

(u-2)(u-6)=0.

So there are two branches:

u=2 \quad\text{or}\quad u=6.

This is where it's easy to stop too soon. You've found $u$ , but the question is about $x$ . Put $x^2+x$ back in for $u$ in both branches:

x^2+x=2 \quad\text{or}\quad x^2+x=6.

Solve both:

\begin{aligned} x^2+x-2&=(x-1)(x+2)=0,\\[1.4em] x^2+x-6&=(x-2)(x+3)=0. \end{aligned}

The complete real solution set is

\{-3,-2,1,2\}.

The substitution did the hard part, but it didn't finish the job. Remember: $u$ is a helper, not the answer. Every value of $u$ has to go back to $x$ .

Try it yourself:

Solve $x^4-13x^2+36=0$ . Since $x^4=(x^2)^2$ , the repeated piece is $x^2$ , so let $u=x^2$ . Factor the quadratic in $u$ , then put $x^2$ back in both branches before you reveal the answer.

Check your understanding:

What are all real solutions to $x^4-13x^2+36=0$ ?

Count distinct roots carefully

Two branches can give you the same root. Distinct solutions means different values, so a root that shows up twice still counts once.

For example,

(x^2-9)(x^2-9x+18)=0

gives

x=-3,\ 3

from the first factor, and

x=3,\ 6

from the second. The root $3$ comes from both branches, so the distinct solution set is

\{-3,3,6\}.

That's three distinct real solutions.

Common mistake:

Counting how many times roots appear instead of how many different values there are. Here that gives $4$ solutions instead of $3$ , and a sum of $9$ instead of $6$ . Write the roots from every branch in one ordered list, and cross out repeats before you count or add them.

Practice problems

Before each problem, read what it asks for and pick your method from that. Then find every root before you answer.

Make one branch add exactly one root

Practice problem

For a constant $k$ , the equation

(x^2-4)^2-k(x^2-4)=0

has exactly three distinct real solutions. What is the value of $k$ ?

Your answer

Calculator loads as you approach

Find k by hand first, using the repeated expression. Then graph the equation with your k to check the count.

Keep the roots from the zero branch

Practice problem

What is the greatest real solution to

(x^2-5x)^2-6(x^2-5x)=0?

Your answer

Calculator loads as you approach

Graph the left side as written and click every x-intercept.

Use a root that shows up twice

Practice problem

For a constant $k$ , the equation

(x^2-9)(x^2-kx+18)=0

has exactly three distinct real solutions. The sum of those three distinct solutions is $6$ . What is the value of $k$ ?

Your answer

Calculator loads as you approach

Work out k from the known roots first. Graphing with your k afterward is a good check.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Look for structure before you expand anything: a product equal to $0$ , a shared factor of $x$ , a repeated expression, or only even powers like $x^4$ and $x^2$ .
For a short all-number equation that asks for a root, graph the polynomial side without expanding and read every $x$ -intercept.
For a constant like $k$ , an answer with letters, an exact boundary, or proof you have every root, use structural algebra.
Split a product into branches only when it equals $0$ . One branch can give more than one $x$ .
Never divide by an expression with $x$ in it before you've kept the case where it's $0$ .
After a substitution, put the full expression back for $u$ in every branch. $u$ is a helper, not the answer.
Combine every root, drop repeats when the question says distinct, and apply words like greatest or positive last.

Related lessons

If the product isn't visible yet, review Factor algebraic expressions. If the equation comes down to an ordinary quadratic, review Solve quadratic equations by factoring.

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162 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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