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Section 5.5 (Easy Factoring) Perfect Square Trinomials & Differences of Squares

Section 5.5 (Easy Factoring) Perfect Square Trinomials & Differences of Squares. Review: The Perfect Square Trinomial Rules ( A + B ) 2 = A 2 + 2 A B + B 2 ( A – B ) 2 = A 2 – 2 A B + B 2 If you see a trinomial that has these patterns, it factors easily:

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Section 5.5 (Easy Factoring) Perfect Square Trinomials & Differences of Squares

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  1. Section 5.5 (Easy Factoring) Perfect Square Trinomials & Differences of Squares • 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 … 5.5

  2. Is the Trinomial a Perfect Square ? • Recall the square of two binomials pattern we used when multiplying: (b)2(±5)2 (b – 5)2 5.5

  3. 1: If necessary, Arrange in Descending Order • Why? Because we will need to check the 1st and 3rd terms, then check the middle term • x2 + 1 – 2x = x2 – 2x + 1 • 25 – 10x + x2 = x2 – 10x + 25 • 6rs + 9s2 +r2 = r2 + 6rs + 9s2 5.5

  4. 2: Remove Any Common Factors (always check this before proceeding) • 3x2 – 15x + 12 = 3(x2 – 5x + 4) = 3(x – 1)(x – 4) • Even when the 1st and 3rd terms are squares • 16x2 + 16x + 4 = 4(4x2 + 4x + 1) = 4(2x + 1)2 • Sometimes a variable factors out • x2y – 10xy + 25y = y(x2 – 10x + 25) = y(x – 5)2 5.5

  5. 3: See if the 1st and 3rd Terms are Squares • Check 36x2 + 84x + 49 • (6x)2 … (7)2 ok, might be! • Check 9x2– 68xy + 121y2 • (3x)2 … (11y)2 ok, might be! • Check 16a2 + 22a + 63 • (4a)2 … (7)(9)no, can’t be! • Ok, let’s see if the middle terms are right 5.5

  6. 4: See if the Middle Term is 2AB • Check 36x2+ 84x + 49 • (6x)2 … (7)2 ok, might be! • 2(6x)(7) = 84x 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 ok, might be! • 2(4a)(-9) = -72a yes, it is (4a – 9)2 5.5

  7. Are These Perfect Square Trinomials? • x2 + 8x + 16 = (x + 4)2 • (x)2 (4)22(x)(4) = 8x yes, it matches • t2–5t + 4 = not a PST… but it factors: (t - 1)(t - 4) • (t)2 (-2)22(t)(-2) = -4t no, it’s not -5t • 25 + y2 + 10y = (y + 5)2 • y2 + 10y + 25 descending order • (y)2 (5)22(y)(5) = 10y yes, it matches • 3x2– 15x + 27 = not a PST • 3(x2– 5x + 9) remove common factor • (x)2 (-3)22(x)(-3) = -6x no, it’s not -5x PST Tests: 1. Descending Order 2. Common Factors 3. 1st and 3rd Terms (A)2 and (B)2 4. Middle Term 2AB or -2AB 5.5

  8. Difference of Squares Binomials • Remember that the middle term disappears? (A + B)(A – B) =A2 - B2 • It’s easy factoring when you find binomials of this patternA2 – B2 = (A + B)(A – B) • Examples: • x2 – 9 = • (x)2 – (3)2 = • (x + 3)(x – 3) • 4t2 – 49 = • (2t)2 – (7)2 = • (2t + 7)(2t – 7) • a2 – 25b2 = two variables squared • (a)2 – (5b)2 = • (a + 5b)(a – 5b) • 18 – 2y4 = constant 1st, variable square 2nd • 2 [ (3)2 – (y2)2 ] = • 2(3 + y2)(3 – y2) 5.5

  9. More Difference of Squares • Examples: • 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) 5.5

  10. What Next? • Section 5.6–Factoring Sums & Differences of Cubes 5.5

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