Factoring Perfect-Square Trinomials and Differences of Squares

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Section 5.5 (Easy Factoring)
Perfect Square Trinomials & Differences of Squares
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Review: The Perfect Square Trinomial Rules
 (A + B)2 = A2 + 2AB + B2
 (A – B) 2 = A2 – 2AB + B2
If you see a trinomial that has these patterns, it factors easily:
 A2 + 2AB + B2 = (A + B)2
 A2 – 2AB + B2 = (A – B)2
Some examples:
 x2 – 2x + 1 = (x – 1)2
 r2 + 6rs + 9s2 = (r + 3s)2
Some trinomials are more difficult to spot,
so we need a reliable procedure
…
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Is the Trinomial a Perfect Square ?
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Recall the square of
two binomials pattern
we used when
multiplying:
(b)2
(±5)2
(b – 5)2
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1: If necessary, Arrange in Descending Order
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Why? Because we will need to check the
1st and 3rd terms, then check the middle term
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x2 + 1 – 2x = x2 – 2x + 1
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25 – 10x + x2 = x2 – 10x + 25
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6rs + 9s2 + r2 = r2 + 6rs + 9s2
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2: Remove Any Common Factors
(always check this before proceeding)
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3x2 – 15x + 12 = 3(x2 – 5x + 4) = 3(x – 1)(x – 4)
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Even when the 1st and 3rd terms are squares
16x2 + 16x + 4 = 4(4x2 + 4x + 1) = 4(2x + 1)2
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Sometimes a variable factors out
x2y – 10xy + 25y = y(x2 – 10x + 25) = y(x – 5)2
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3: See if the 1st and 3rd Terms are Squares
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Check 36x2 + 84x + 49
(6x)2 …
(7)2
Check 9x2 – 68xy + 121y2
(3x)2 … (11y)2
ok, might be!
Check 16a2 + 22a + 63
(4a)2 … (7)(9)
no, can’t be!
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ok, might be!
Ok, let’s see if the middle terms are right
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4: See if the Middle Term is 2AB
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Check 36x2 + 84x + 49
(6x)2 …
(7)2
2(6x)(7) = 84x
ok, might be!
yes, it is (6x + 7)2
Check 9x2 – 68xy + 121y2
(3x)2 … (-11y)2
ok, might be!
2(3x)(-11y) = -66xy ≠ -68xy no, not a PST
Check 16a2 – 72a + 81
(4a)2 …
(-9)2
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2(4a)(-9) = -72a
ok, might be!
yes, it is (4a – 9)2
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Are These Perfect Square Trinomials?
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x2 + 8x + 16 = (x + 4)2
(x)2
(4)2 2(x)(4) = 8x
yes, it matches
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t2 – 5t + 4 = not a PST… but it factors: (t - 1)(t - 4)
(t)2
(-2)2 2(t)(-2) = -4t no, it’s not -5t
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PST Tests:
1. Descending Order
2. Common Factors
3. 1st and 3rd Terms
(A)2 and (B)2
4. Middle Term
2AB or -2AB
25 + y2 + 10y = (y + 5)2
y2 + 10y + 25
descending order
(y)2
(5)2 2(y)(5) = 10y yes, it matches
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3x2 – 15x + 27 = not a PST
3(x2 – 5x + 9)
remove common factor
(x)2
(-3)2 2(x)(-3) = -6x no, it’s not -5x
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Difference of Squares Binomials
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Remember that the middle term disappears? (A + B)(A – B) = A2 - B2
It’s easy factoring when you find binomials of this pattern
A2 – B2 = (A + B)(A – B)
Examples:
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x2 – 9 =
(x)2 – (3)2 =
(x + 3)(x – 3)
4t2 – 49 =
(2t)2 – (7)2 =
(2t + 7)(2t – 7)
a2 – 25b2 =
(a)2 – (5b)2 =
(a + 5b)(a – 5b)
18 – 2y4 =
2 [ (3)2 – (y2)2 ] =
2(3 + y2)(3 – y2)
two variables squared
constant 1st, variable square 2nd
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More Difference of Squares
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Examples:
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x2 – 1/9 =
perfect square fractions
(x)2 – (⅓)2 =
(x + ⅓)(x – ⅓)
18x2 – 50x4 =
common factors must be removed
2x2[ 9 – 25x2 ] =
2x2[ (3)2 – (5x)2 ] =
2x2(3 + 5x)(3 – 5x)
p8 – 1 =
factor completely
(p4)2 – (1)2 =
(p4 + 1)(p4 – 1)
another difference of 2 squares
(p4 + 1)(p2 + 1)(p2 – 1) and another
(p4 + 1)(p2 + 1)(p + 1)(p
– 1)
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What Next?
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Section 5.6 –
Factoring Sums &
Differences of Cubes
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