1.3 Algebraic Expressions
1.3.1 Learning Objectives
By the end of this section, you should be able to:
- identify polynomials and determine their degrees;
- add, subtract, and multiply polynomials;
- use special-product formulas; and
- factor expressions completely.
1.3.2 Algebraic Expressions and Polynomials
A variable represents a number from a specified set. Combining variables and numbers using arithmetic operations, powers, and roots produces an algebraic expression.
A polynomial in \(x\) has the form
\[ a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0, \]
where the exponents are nonnegative integers. Its degree is the greatest exponent with a nonzero coefficient. A one-term polynomial is a monomial, a two-term polynomial is a binomial, and a three-term polynomial is a trinomial.
Terms are like terms when their variable factors are identical. Like terms can be combined using the distributive property:
\[ 4x^2-7x^2=(4-7)x^2=-3x^2. \]
Example 1.12 Polynomial Operations
\[ (3x^2-2x+5)-(x^2+4x-1) =\boxed{2x^2-6x+6}. \]
Exercise 1.17 State the degree of \(7x^5-2x^3+x-8\).
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Answer: \(5\)
Exercise 1.18 Simplify \((5x^2+x)-(2x^2-4x)\).
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Answer: \(3x^2+5x\)
1.3.3 Multiplying Polynomials and Special Products
To multiply polynomials, distribute every term in one factor across every term in the other.
Example 1.13 Multiplying
\[ (2x-3)(x+5) =2x^2+10x-3x-15 =\boxed{2x^2+7x-15}. \]
Frequently used patterns include
\[ (A+B)^2=A^2+2AB+B^2, \]
\[ (A-B)^2=A^2-2AB+B^2, \]
\[ (A-B)(A+B)=A^2-B^2. \]
Example 1.14 Special Products
\[ (3x+4)^2 =9x^2+24x+16. \]
Exercise 1.19 Expand \((x-6)^2\).
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Answer: \(x^2-12x+36\)
1.3.4 Factoring Polynomials
Factoring reverses multiplication. Always look first for a greatest common factor. Other useful patterns are
\[ A^2-B^2=(A-B)(A+B), \]
\[ A^3-B^3=(A-B)(A^2+AB+B^2), \]
\[ A^3+B^3=(A+B)(A^2-AB+B^2). \]
A trinomial \(x^2+bx+c\) factors as \((x+r)(x+s)\) when \(r+s=b\) and \(rs=c\). Some four-term polynomials can be factored by grouping pairs of terms.
Example 1.15 Factoring a Trinomial
\[ x^2-x-20=\boxed{(x-5)(x+4)} \]
because \((-5)(4)=-20\) and \(-5+4=-1\).
Exercise 1.20 Factor \(x^2+9x+20\).
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Answer: \((x+4)(x+5)\)
Example 1.16 Factoring Completely
\[ 6x^3-24x =6x(x^2-4) =\boxed{6x(x-2)(x+2)}. \]
Exercise 1.21 Factor \(8x^3+12x^2\).
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Answer: \(4x^2(2x+3)\)
Exercise 1.22 Factor \(25y^2-16\).
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Answer: \((5y-4)(5y+4)\)
1.3.5 Conceptual Takeaways
- Polynomial terms have nonnegative integer exponents.
- Like terms have exactly the same variable factors.
- Multiplication expands expressions; factoring reverses that process.
- Special products provide efficient expansion and factoring patterns.
- Complete factoring often requires more than one step.
1.3.6 Skills You Should Be Able to Do
- Identify polynomial terms, type, and degree.
- Add, subtract, and multiply polynomials.
- Apply special product formulas.
- Factor out a greatest common factor.
- Factor trinomials, differences of squares, sums and differences of cubes, and grouped expressions.
1.3.7 Practice Problems with Solutions
Identify the degree and number of terms in \(4x^6-3x^2+9\).
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The polynomial has three terms and degree \(\boxed{6}\).
Simplify \((2x^3-5x+1)+(x^3+7x-6)\).
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\[ 2x^3+x^3-5x+7x+1-6 =\boxed{3x^3+2x-5}. \]
Expand \((3x-2)(x+4)\).
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\[ (3x-2)(x+4)=\boxed{3x^2+10x-8}. \]
Expand \((2a+5)^2\).
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\[ (2a+5)^2=4a^2+20a+25. \]
Factor \(15x^4-10x^3\).
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\[ 15x^4-10x^3=\boxed{5x^3(3x-2)}. \]
Factor \(x^2-11x+24\).
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\[ x^2-11x+24=\boxed{(x-3)(x-8)}. \]
Factor \(9y^2-49\).
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\[ 9y^2-49=\boxed{(3y-7)(3y+7)}. \]
Factor \(2x^3+6x^2-x-3\) completely.
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\[ 2x^3+6x^2-x-3 =2x^2(x+3)-1(x+3) \] \[ =(x+3)(2x^2-1). \]
Thus the factorization over the integers is \(\boxed{(x+3)(2x^2-1)}\).