Polynomials, like terms, and standard form

Key idea

A polynomial in \(x\) is built from terms like \(a x^n\), where \(a\) is a number and the exponent \(n\) is a whole number (\(0,1,2,\dots\)). To simplify, you combine like terms (same variable and same exponent). In standard form, terms are written in descending powers of \(x\).

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Summary

• Polynomials use whole-number exponents of the variable.
• Combine like terms to simplify, then write answers in standard form.

Add and subtract polynomials

Key idea

To add polynomials, add the coefficients of like terms. To subtract polynomials, distribute the minus sign across every term in the parentheses: \[ (3x^2+2)-(x^2+1)=3x^2+2-x^2-1. \] Then combine like terms.

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Summary

• Add: combine like terms.
• Subtract: distribute the negative sign, then combine like terms.

Multiply polynomials with distribution and exponent rules

Key idea

When multiplying powers with the same base, add exponents: \[ x^a \cdot x^b = x^{a+b}. \] To multiply polynomials, distribute each term: \[ (a+b)(c+d)=ac+ad+bc+bd. \]

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Summary

• Exponent rule: \(x^a \cdot x^b = x^{a+b}\).
• Use distribution (FOIL) to multiply binomials and larger polynomials.

Special products that speed up expansion

Key idea

Some products show up so often that it's worth memorizing the patterns: \[ (a+b)^2 = a^2 + 2ab + b^2,\quad (a-b)^2 = a^2 - 2ab + b^2. \] These patterns help you expand faster and also recognize what to factor later.

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Summary

• Binomial square: \((a+b)^2=a^2+2ab+b^2\).
• These patterns reduce mistakes and make factoring easier later.

Factor polynomials: patterns and grouping

Key idea

Factoring rewrites a polynomial as a product. A reliable order is: (1) GCF → (2) special patterns → (3) grouping. A classic pattern is the difference of squares: \[ a^2-b^2=(a-b)(a+b). \]

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Summary

• Difference of squares: \(a^2-b^2=(a-b)(a+b)\).
• Grouping is useful when you have four terms and can factor a common binomial.

Polynomial division, synthetic division, and the remainder theorem

Key idea

When you divide a polynomial \(f(x)\) by a linear factor \((x-a)\), you get: \[ f(x)=(x-a)\,q(x)+r, \] where \(q(x)\) is the quotient and \(r\) is the remainder (a constant). The remainder theorem says: \[ r=f(a). \]

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Summary

• Dividing by \(x-a\) gives \(f(x)=(x-a)q(x)+r\).
• Remainder theorem: the remainder is \(r=f(a)\).

Why polynomial fundamentals matter

Where polynomials show up

• Algebra and functions: polynomial graphs, intercepts, and end behavior.
• Geometry: area and volume formulas expand into polynomials.
• Science and engineering: approximations and models often use polynomial expressions.
• Computing and data: curve fitting and interpolation use polynomials.

Worked example: evaluate a polynomial

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Final recap

• Polynomials use whole-number exponents and combine like terms to simplify.
• Add/subtract: distribute negatives, then combine like terms.
• Multiply: use distribution and exponent rules; learn special product patterns.
• Factor: start with GCF, then patterns (difference of squares), then grouping.
• Division: \(f(x)=(x-a)q(x)+r\) and the remainder is \(f(a)\).