Intro to Polynomials Answer Key
Teacher Answer Key Algebra 1 • Unit 7
Intro to Polynomials Notes
Course: Foundations of Algebra
Topic: Vocabulary & Standard Form
Word
Definition
Examples & Solutions
Term
Numbers and variables that are multiplied together.
\(3x,\; 10,\; 5x^2,\; x^4\)
Expression
Two or more terms (connected by \(+\) or \(-\)).
\(3x + 7\)
Polynomial
An expression with many terms.
\(5x^3 + 3x^2 - 10x + 2\)
Degree
The highest / greatest exponent in a polynomial.
\(7x^{\mathbf{5}} + 6x^3 + 2x\) Degree = 5
Standard Form
When a polynomial is arranged from greatest exponent / degree to least exponent / degree.
\(10 - 4x^3 + 2x^5 - 8x^2\) Rewrite: \(2x^5 - 4x^3 - 8x^2 + 10\)
Leading Coefficient
The number written in front of the variable with the highest / greatest exponent.
\(\mathbf{8}x^3 + 5x^2 - x + 7\) Leading Coeff = 8
Examples & Practice (Questions 1 - 6) Detailed Student Solutions
1. How many terms? Count
\(9x^4 + 3x^3 - 2x^2 - 5x + 2\)
Terms: \(9x^4, +3x^3, -2x^2, -5x, +2\)
Answer: 5 terms
2. What is the degree? Highest Exp
\(3x^6 - 10x^3\)
Compare exponents: 6 vs. 3
Degree = 6
3. Standard form: Order
\(16 - 10x^3 + 5x^4 - 2x^2\)
Exponents in order: 4, 3, 2, 0
\(5x^4 - 10x^3 - 2x^2 + 16\)
4. Leading coefficient? Coeff
\(-7x^3 + 6x^2 - 5\)
Degree is 3; front number on \(x^3\)
Answer: \(-7\)
5. Standard form: Order
\(4x^4 - 10x^7 + 3x^2 + 1\)
Exponents: 7, 4, 2, 0 (keep signs!)
\(-10x^7 + 4x^4 + 3x^2 + 1\)
6. What is the degree? Highest Exp
\(5x^3 - 2x^2 + 28\)
Highest exponent on any variable
Degree = 3
Tip
Common Misconceptions: In #5, remind students that the negative sign stays with \(10x^7\) as its coefficient \(-10\). In #1, remind students that constants count as individual terms.
Intro to Polynomials Notes • Answer Key Page 1 of 3
Teacher Answer Key Algebra 1 • Unit 7
Combining Like Terms
Method: Grouping Similar Parts
Rule: Add/Subtract Coefficients
Definition: Combining like terms is a way of simplifying algebraic expressions by grouping similar parts together. When we combine like terms, we add or subtract their coefficients.
Expression with 4 terms:
3a + 4b + 2a - 2b
Notice the \(+\) or \(-\) sign in front of a term belongs to that term.
Combining "\(a\)" terms:
\(3a\) is 3 groups of \(a \to +a +a +a\)
\(2a\) is 2 groups of \(a \to +a +a\)
Both positive \(\to 3a + 2a = 5a\)
Combining "\(b\)" terms (Zero Pairs):
\(+4b = +b +b +b +b\)
\(-2b = -b -b\) (cancels two \(+b\)'s)
Result: \(4b - 2b = 2b\)
Final Simplified Form: \(5a + 2b\) (Cannot combine further because \(5a\) and \(2b\) are not like terms)
Step-by-Step Problem Solutions
Example 1: Simplify Single Variable
\(6a - 3a + 5a + 5\)
Group \(a\) terms: \((6a - 3a + 5a) + 5\)
Combine coefficients: \((6 - 3 + 5)a = 8a\)
Constant: \(+5\) has no like term
Answer: \(8a + 5\)
Example 2: Simplify Two Variables
\(7a + 4b + 2a + 3b\)
Group \(a\)'s: \(7a + 2a = 9a\)
Group \(b\)'s: \(4b + 3b = 7b\)
Alphabetical order is standard form.
Answer: \(9a + 7b\)
3. Simplify & Standard Form Standard Form
\(5x^3 + 12 + 8x^3 - 7\)
Group \(x^3\) terms: \(5x^3 + 8x^3 = 13x^3\)
Group constants: \(12 - 7 = +5\)
Arranged descending: degree 3, degree 0.
Answer: \(13x^3 + 5\)
4. Simplify & Standard Form Standard Form
\(6x^2 + 10x - 4x^2 + 12x\)
Group \(x^2\) terms: \(6x^2 - 4x^2 = 2x^2\)
Group \(x\) terms: \(10x + 12x = 22x\)
Standard order: \(x^2\) term before \(x\) term.
Answer: \(2x^2 + 22x\)
Note
Guiding Question: "Can we combine \(2x^2\) and \(22x\)?" Emphasize that although both share base \(x\), they have different exponents and therefore are not like terms.
Intro to Polynomials Notes • Answer Key Page 2 of 3
Teacher Answer Key Algebra 1 • Unit 7
Multiplying Variables & Applications
Skills: Perimeter, Exponents & Area
Key Law: \(x^a \cdot x^b = x^{a+b}\)
5. Perimeter of Triangle Application
Sum of All Sides
(5x + 13) (8x - 2) (3x + 10) Diagram not to exact scale
Formula: \(\text{Perimeter} = \text{Side}_1 + \text{Side}_2 + \text{Side}_3\)
\(P = (5x + 13) + (8x - 2) + (3x + 10)\)
Combine \(x\) terms: \(5x + 8x + 3x = 16x\)
Combine constants: \(13 - 2 + 10 = +21\)
Standard Form \(16x + 21\)
Multiplying Variables Concepts Base, Exponent & Product Rule
A variable is a symbol for a number we don't know yet. It is usually a letter like \(x\) or \(y\).
An exponent says how many times to use the variable in a multiplication.
Expanded Form & Product Rule
\(z^3 =\) \(z \cdot z \cdot z\)
\(x^5 =\) \(x \cdot x \cdot x \cdot x \cdot x\)
\(y \cdot y \cdot y \cdot y =\) \(y^4\)
\(x^2 \cdot x =\) \(x^3\)
Distributive Property with Variables
\(z(3 + z) =\) \(3z + z^2\) (\(z^2 + 3z\))
\(x(5x + 2) =\) \(5x^2 + 2x\)
\(y^2(7y + 3) =\) \(7y^3 + 3y^2\)
\(z^3(z^2 - 3z + 2y) =\) \(z^5 - 3z^4 + 2yz^3\)
Area of the Rectangle Problem
\(\text{Area} = \text{length} \times \text{width}\)
5x 5x - 4 Dimensions: \(5x\) by \((5x - 4)\)
Set up multiplication: \(A = 5x(5x - 4)\)
Distribute \(5x\): \((5x \cdot 5x) - (5x \cdot 4) = 25x^2 - 20x\)
Final Answer: \(25x^2 - 20x\)
Alert
Watch for exponent rules: When multiplying \(x \cdot x\), remind students that invisible exponent 1 is present on each factor (\(x^1 \cdot x^1 = x^2\)). Do not confuse with addition (\(x + x = 2x\)).
Intro to Polynomials Notes • Answer Key Page 3 of 3
