Topic 1.2 · Unit 1 · CA CCSS A-APR.1
Adding, subtracting and multiplying polynomials
Adding and subtracting polynomials is combining like terms with one trap, the subtraction sign. Multiplying is the distributive property applied until nothing is left in brackets.
- Adding: combine like terms. Only identical variable parts combine, so \( 3x^2 \) and \( 3x \) stay separate.
- Subtracting: distribute the minus to every term in the second polynomial, then add. This single step accounts for most errors in the topic.
- Multiplying a monomial by a polynomial: distribute to each term, multiplying numbers and adding exponents.
- Multiplying two binomials: every term in the first multiplies every term in the second, four products for two binomials. FOIL is a name for this order, not a separate rule, and it only works for binomial times binomial.
- Two patterns worth recognizing instantly: \( (a+b)(a-b) = a^2 - b^2 \), the middle terms cancel. And \( (a+b)^2 = a^2 + 2ab + b^2 \), not \( a^2 + b^2 \).
Where marks are lost: \( (x+3)^2 \) is not \( x^2 + 9 \). Squaring a binomial means multiplying it by itself, which produces a middle term: \( x^2 + 6x + 9 \). Test with \( x = 1 \): \( 4^2 = 16 \), and \( 1 + 6 + 9 = 16 \) ✓, while \( 1 + 9 = 10 \) ✗.
Example 1: subtraction. Simplify \( (5x^2 - 3x + 4) - (2x^2 + x - 7) \).
Distribute the minus across all three terms of the second: \( -2x^2 - x + 7 \). Now combine: \( 5x^2 - 2x^2 = 3x^2 \); \( -3x - x = -4x \); \( 4 + 7 = 11 \).
Result \( 3x^2 - 4x + 11 \). The \( +7 \) is where marks go; it was \( -7 \) inside the bracket.
Example 2: two binomials. Expand \( (2x + 3)(x - 5) \).
Four products: \( 2x \cdot x = 2x^2 \); \( 2x \cdot -5 = -10x \); \( 3 \cdot x = 3x \); \( 3 \cdot -5 = -15 \).
Combining the middle terms: \( 2x^2 - 7x - 15 \).
Example 3: the difference of squares pattern. Expand \( (x + 6)(x - 6) \).
The middle products are \( -6x \) and \( +6x \), which cancel, leaving \( x^2 - 36 \). Recognizing this pattern going forwards makes factoring it backwards in the next topic much faster.
1–5 add and subtract, 6–12 multiply, 13–15 use the patterns and diagnosis.
- Simplify \( (3x + 5) + (2x - 8) \).
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\( 5x - 3 \). \( 5x - 3 \)
- Simplify \( (4x^2 + x) + (x^2 - 3x) \).
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\( 5x^2 - 2x \). \( 5x^2 - 2x \)
- Simplify \( (7x - 2) - (3x + 6) \).
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Distribute the minus: \( -3x - 6 \). Then \( 4x - 8 \). \( 4x - 8 \)
- Simplify \( (x^2 - 4x + 1) - (2x^2 - x - 5) \).
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Second becomes \( -2x^2 + x + 5 \). Combining: \( -x^2 - 3x + 6 \). \( -x^2 - 3x + 6 \)
- Simplify \( (6x^2 + 2x - 9) - (6x^2 - 2x + 9) \).
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Second becomes \( -6x^2 + 2x - 9 \). The \( x^2 \) terms cancel: \( 4x - 18 \). \( 4x - 18 \)
- Expand \( 3x(x + 4) \).
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\( 3x^2 + 12x \)
- Expand \( -2x(3x - 5) \).
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\( -6x^2 + 10x \). The second sign flips. \( -6x^2 + 10x \)
- Expand \( (x + 2)(x + 7) \).
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\( x^2 + 7x + 2x + 14 \). \( x^2 + 9x + 14 \)
- Expand \( (x - 3)(x + 8) \).
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\( x^2 + 8x - 3x - 24 \). \( x^2 + 5x - 24 \)
- Expand \( (2x + 1)(3x - 4) \).
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\( 6x^2 - 8x + 3x - 4 \). \( 6x^2 - 5x - 4 \)
- Expand \( (x - 5)(x + 5) \).
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Middle terms cancel. \( x^2 - 25 \)
- Expand \( (x + 4)^2 \).
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\( (x+4)(x+4) = x^2 + 4x + 4x + 16 \). \( x^2 + 8x + 16 \)
- Expand \( (3x - 2)^2 \).
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\( 9x^2 - 6x - 6x + 4 \). \( 9x^2 - 12x + 4 \)
- A student writes \( (x + 5)^2 = x^2 + 25 \). Find the error.
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They squared each term instead of multiplying the binomial by itself, losing the middle term: \( x^2 + 10x + 25 \). Test \( x = 1 \): \( 6^2 = 36 \), and \( 1 + 10 + 25 = 36 \) ✓, while their answer gives 26. \( x^2 + 10x + 25 \)
- A student simplifies \( (5x - 1) - (2x - 4) \) to \( 3x - 5 \). Find the error.
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They subtracted the \( 2x \) correctly but kept the \( -4 \) negative. Distributing the minus gives \( -2x + 4 \), so the answer is \( 3x + 3 \). \( 3x + 3 \)