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Grade 11 Precalculus · Lesson 1.3

Algebraic Expressions

Read the structure first, then choose the rule. Move through operations, special products, factoring, rational exponents, substitution, identities, and mastery practice.
9 learning scenesInteractiveSame lesson material

Complete lesson · Operations, special products, factoring, rational exponents, substitution, identities, and mastery practice.

Polynomial Language

Read the structure first: terms, coefficients, leading coefficient, constant term, degree, and standard form.

f(x)=anxn+an−1xn−1+⋯+a1x+a0f(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0
ExplanationTerm

A part separated by + or −.

ExplanationCoefficient

The numerical factor of a variable term.

ExplanationDegree

The highest exponent after simplifying and writing in standard form.

Read the polynomial
P(x)=−4x5+3x3−7x+9P(x)=-4x^5+3x^3-7x+9
Degree: 5
Leading coefficient: -4
Constant term: 9
Number of terms: 4

Quick Check

Check the idea before moving on.

What is the degree after the polynomial is written in standard form?
4−3x2+7x5−x4-3x^2+7x^5-x
Common mistake. The degree is the highest exponent after simplifying and writing the polynomial in standard form.

Exponent Toolkit

Recognize the exponent structure first, then apply the matching rule.

aman=am+na^m a^n=a^{m+n}
aman=am−n\frac{a^m}{a^n}=a^{m-n}
(am)n=amn(a^m)^n=a^{mn}
am/n=amn=(an)ma^{m/n}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m
a−m=1ama^{-m}=\frac1{a^m}
a0=1(a≠0)a^0=1\quad(a\ne0)
Interactive worked example · choose the exponent rule
x3/2x1/2x^{3/2}x^{1/2}

Name the rule before revealing the calculation.

Your turn · Recognize the structure

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interactive worked example · power of a power
(x2/3)3\left(x^{2/3}\right)^3

Decide whether the exponents should be added or multiplied.

Your turn · Recognize the structure

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Negative rational exponent
x−3/2=1x3/2=1(x)3x^{-3/2}=\frac{1}{x^{3/2}}=\frac{1}{(\sqrt{x})^3}

Your turn

Rewrite using positive exponents. x−5/3x^{-5/3}
Common mistake. When multiplying powers with the same base, add exponents. Multiply exponents only when a power is raised to another power.

Add & Subtract Polynomial Expressions

Combine like terms only. The variable part and its exponent must match.

axn+bxn=(a+b)xnax^n+bx^n=(a+b)x^n
Addition
(3x2−5x+4)+(2x2+7x−9)(3x^2-5x+4)+(2x^2+7x-9)=(3x2+2x2)+(−5x+7x)+(4−9)=(3x^2+2x^2)+(-5x+7x)+(4-9)=5x2+2x−5=5x^2+2x-5
Subtraction — distribute the negative first
(4x3−2x+7)−(x3+5x−3)(4x^3-2x+7)-(x^3+5x-3)=4x3−2x+7−x3−5x+3=4x^3-2x+7-x^3-5x+3=3x3−7x+10=3x^3-7x+10

Quick Check

Check the idea before moving on.

What is the FIRST step?
(5x2+3x−1)−(2x2−4x+6)(5x^2+3x-1)-(2x^2-4x+6)
Common mistake. A minus sign before parentheses changes every sign inside the parentheses.

Multiplying Algebraic Expressions

Start with the distributive property. Every term inside the parentheses must be multiplied.

a(b+c+d)=ab+ac+ada(b+c+d)=ab+ac+ad
Interactive worked example · monomial × polynomial
2x2y(3xy−x3+5)2x^2y(3xy-x^3+5)

Distribute the monomial to every term and handle coefficients and exponents separately.

Your turn · First term

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interactive worked example · binomial × binomial
(2x+1)(3x−5)(2x+1)(3x-5)

Make sure every term in the first binomial multiplies every term in the second.

Your turn · Distribute

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Quick Check

Check the idea before moving on.

What must happen before combining like terms?
(x+4)(2x−3)(x+4)(2x-3)
Treat a repeated expression as one object
(x+y−1)(x+y+1)(x+y-1)(x+y+1)A=x+yA=x+y(A−1)(A+1)=A2−1(A-1)(A+1)=A^2-1=x2+2xy+y2−1=x^2+2xy+y^2-1

Special Products

Recognize the pattern first. Identify A and B, then apply the matching rule.

Explanation
(A−B)(A+B)=A2−B2(A-B)(A+B)=A^2-B^2
Explanation
(A+B)2=A2+2AB+B2(A+B)^2=A^2+2AB+B^2
Explanation
(A−B)2=A2−2AB+B2(A-B)^2=A^2-2AB+B^2
Explanation
(A+B)3=A3+3A2B+3AB2+B3(A+B)^3=A^3+3A^2B+3AB^2+B^3
Interactive worked example · recognize conjugates
(3x−4)(3x+4)(3x-4)(3x+4)

Identify A and B before using the shortcut.

Your turn · Pattern

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Quick Check

Check the idea before moving on.

Which pattern applies?
(5x−2y)(5x+2y)(5x-2y)(5x+2y)
Interactive worked example · square of a sum
(3x+4y)2(3x+4y)^2

Predict the three terms.

Your turn · Identify

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Your turn

Find only the middle term. (2x+3y)2(2x+3y)^2
Square of a difference
(2x−5)2(2x-5)^2A=2x,B=5A=2x,\quad B=5=4x2−20x+25=4x^2-20x+25

Quick Check

Check the idea before moving on.

What sign must the middle term have?
(4x−5)2(4x-5)^2
(A+B)3=A3+3A2B+3AB2+B3(A+B)^3=A^3+3A^2B+3AB^2+B^3
Cube of a sum
(x+2)3(x+2)^3A=x,B=2A=x,\quad B=2=x3+6x2+12x+8=x^3+6x^2+12x+8

Quick Check

Check the idea before moving on.

How many terms should the expansion contain?
(x+2)3(x+2)^3
(A−B)3=A3−3A2B+3AB2−B3(A-B)^3=A^3-3A^2B+3AB^2-B^3
Cube of a difference
(x−2)3(x-2)^3A=x,B=2A=x,\quad B=2=x3−6x2+12x−8=x^3-6x^2+12x-8

Mr. Azzam's Factoring System

The routine

FIRST: Always check for a Greatest Common Factor (GCF). Then count terms, choose the pattern, and finally ask: Can I factor again?

Check the sign, numerical factor, variables using the smallest exponent, and whether a complete expression is common.

GCF first
12x3y2−18x2y312x^3y^2-18x^2y^3=6x2y2(2x−3y)=6x^2y^2(2x-3y)

Your turn

What is the FIRST step? 18x4y2−24x3y518x^4y^2-24x^3y^5
A complete expression can be the GCF
y(y−6)+9(y−6)y(y-6)+9(y-6)=(y−6)(y+9)=(y-6)(y+9)

Quick Check

Check the idea before moving on.

Identify the complete common expression.
m(m+4)−7(m+4)m(m+4)-7(m+4)
After the GCF, count the terms: two terms → special patterns; three terms → trinomial patterns; four terms → test grouping.
Two terms — difference of squares
9x2−259x^2-25=(3x)2−52=(3x)^2-5^2=(3x−5)(3x+5)=(3x-5)(3x+5)
Two terms — sum of cubes
x3+8=x3+23x^3+8=x^3+2^3=(x+2)(x2−2x+4)=(x+2)(x^2-2x+4)
Two terms — difference of cubes
8x3−27=(2x)3−338x^3-27=(2x)^3-3^3=(2x−3)(4x2+6x+9)=(2x-3)(4x^2+6x+9)

Quick Check

Check the idea before moving on.

After checking the GCF, what pattern do you see?
25x2−4925x^2-49
ExplanationSimple trinomial

x2+bx+cx^2+bx+c

ExplanationGeneral trinomial

ax2+bx+c, a≠1ax^2+bx+c,\ a\ne1

ExplanationPerfect square

First and last terms are squares; middle term is ±2AB.

Simple trinomial
x2+7x+12x^2+7x+123+4=7,3⋅4=123+4=7,\quad3\cdot4=12=(x+3)(x+4)=(x+3)(x+4)
General trinomial — Trial and Error
6x2+11x+36x^2+11x+3=(3x+1)(2x+3)=(3x+1)(2x+3)6x2+9x+2x+3=6x2+11x+36x^2+9x+2x+3=6x^2+11x+3
Perfect-square trinomial
x2−10x+25x^2-10x+25=x2−2(x)(5)+52=x^2-2(x)(5)+5^2=(x−5)2=(x-5)^2

Quick Check

Check the idea before moving on.

Which description should guide your method?
6x2+13x+66x^2+13x+6
Four terms — factor by grouping
x3+3x2+2x+6x^3+3x^2+2x+6=x2(x+3)+2(x+3)=x^2(x+3)+2(x+3)=(x+3)(x2+2)=(x+3)(x^2+2)

Quick Check

Check the idea before moving on.

What is the useful first move?
2x3+6x2+5x+152x^3+6x^2+5x+15
Factor completely
3x3−27x3x^3-27x=3x(x2−9)=3x(x^2-9)=3x(x−3)(x+3)=3x(x-3)(x+3)

Your turn

What should you ask next? 4x(x2−16)4x(x^2-16)
Common mistake. Do not stop after taking out the GCF. Always check every remaining factor again.
Interactive lab

Factoring Decision Lab

Follow the routine: GCF → count terms → recognize the pattern → factor completely over the real numbers.

Your expression

12x3−48x12x^3-48x

Checkpoint 1 of 3. Choose an answer, check it, then continue. Reset keeps this case.

Worked reference · all steps

1. Is there a nontrivial greatest common factor?

Take out the GCF: 12x12x

2. How many terms are in the original expression?

The original expression has 2 terms.

3. Which factoring route is appropriate?

Take out the GCF: 12x12x12x(x2−4)12x(x^2-4)Factor again: 12x(x−2)(x+2)12x(x-2)(x+2)
Is there a nontrivial greatest common factor?

Advanced Factoring

When the same expression repeats, treat it as one object first.

Interactive worked example · repeated expression substitution
(x2+1)2+2(x2+1)−3(x^2+1)^2+2(x^2+1)-3

Spot the repeated object before doing any expansion.

Your turn · Choose the object

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Quick Check

Check the idea before moving on.

Which substitution reveals a quadratic?
x4−7x2+12x^4-7x^2+12

With rational or negative exponents, factor out the lowest power first.

Interactive worked example · lowest rational power
x3/2+3x1/2x^{3/2}+3x^{1/2}

Compare the exponents before choosing the GCF.

Your turn · Compare powers

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Interactive worked example · negative rational powers
x−1/2+3x−3/2x^{-1/2}+3x^{-3/2}

The more negative exponent is the lower power.

Your turn · Compare powers

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Quick Check

Check the idea before moving on.

Which power should be factored out?
x7/4−5x3/4x^{7/4}-5x^{3/4}
Interactive worked example · factor completely
2x3−18x2x^3-18x

Follow the routine and do not stop after the first step.

Your turn · GCF first

Work this step on paper first. The solution stays hidden until you confirm you are ready.

Simplify a large algebraic expression
4−2(2x−5)(x+3)+3(x−2)24-2(2x-5)(x+3)+3(x-2)^2=4−2(2x2+x−15)+3(x2−4x+4)=4-2(2x^2+x-15)+3(x^2-4x+4)=−x2−14x+46=-x^2-14x+46

Algebraic Identities & Proof

To prove an identity, work on one side using algebraic rules until it becomes the other side.

Prove using expansion
(x+y)2−(x−y)2=4xy(x+y)^2-(x-y)^2=4xyx2+2xy+y2−(x2−2xy+y2)x^2+2xy+y^2-(x^2-2xy+y^2)=4xy=4xy
Prove using factoring
x2−y2=(x−y)(x+y)x^2-y^2=(x-y)(x+y)(x−y)(x+y)=x2−y2(x-y)(x+y)=x^2-y^2

Your turn

What is a valid way to prove an identity?
Common mistake. Do not assume the identity is true as a step. Show the algebra that transforms one side into the other.

Mastery Checkpoint

Rate each skill before Practice. Your selections route you toward the skills that need more work.

Lesson Mastery Checkpoint

Rate each skill. This is only the starting signal; your actual Practice and Assessment performance will update the plan.

Polynomial language

Exponent rules

Adding and subtracting expressions

Multiplying expressions

Special products

Factoring system

Advanced factoring and substitution

Algebraic identities

Rate all skills to prepare your personalized practice path.

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