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Intermediate Algebra

OPENSTAX

$Math Studyset 91影视 Explanations$ Math

OER 2017 Edition 路 1346 pages

ISBN: 9780998625720

Chapter 6: Q. 387 (page 635)

In the following exercise, factor completely using the perfect square trinomials pattern.
$75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2}$

Short Answer

Expert verified

The factorisation of the given polynomial is:

$75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2} = 3 u^{2} {(5 u - v)}^{2}$

Step by step solution

01

Step 1. Given information

The given polynomial is:

$75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2}$

02

Step 2. Factorise the polynomial

We know that,

${(a + b)}^{2} = a^{2} - 2 a b + b^{2}$

For the polynomial $75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2}$ , taking out $3 u^{2}$ as a common, we get:

localid="1644657545562" $3 u^{2} (25 u^{2} - 10 u v + 3 v^{2})$

Now,

localid="1644657824561" $3 u^{2} (25 u^{2} - 10 u v + v^{2}) 3 u^{2} [{(5 u)}^{2} - 2 脳 5 u 脳 v + v^{2}] 3 u^{2} {(5 u - v)}^{2} [u \sin g {(a - b)}^{2} = a^{2} - 2 a b + b^{2}]$

The factorisation of the given polynomial is:

localid="1644657866268" $75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2} = 3 u^{2} {(5 u - v)}^{2}$

03

Step 3. Check the answer

Multiplying the factors we get:

$75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2} = 3 u^{2} {(5 u - v)}^{2} 75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2} = 3 u^{2} (5 u - v) (5 u - v) 75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2} = 3 u^{2} (25 u^{2} - 10 u v + v^{2}) 75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2} = 75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2}$

This is true.

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