OPERATIONS WITH POLYNOMIALS PRACTICE SAT

Problem 1 :

(3x + 2y)²

If the expression above can be written as ax² + bxy + cy², where a, b and c are constants, what is the value of a + b + c ?

Solution :

Expanding (3x + 2y)² using the algebraic identity (a+b)², we get 

= (3x)² + 2(3x)(2y) + (2y)²

= 9x² + 12xy + 4y²

a = 9, b = 12 and c = 4

a + b + c = 9 + 12 + 4

a + b + c = 25

Problem 2 :

Which of the following expression is not equivalent to

ab (c + d) ?

(a)  abd + abc     (b)  ab(d + c)     (c)  ab (c + d)   (d) abc + bd

Solution :

ab (c + d)

Using distributive property, we get

= abc + abd

So, option c is not equivalent to the given.

Problem 3 :

x² + kx + 9 = (x + a)²

In the equation above, k and a are positive constants. If the equation is true for all values of x, what us the value of k ?

Solution :

x² + kx + 9 = (x + a)²

x² + kx + 9 = x² + 2ax + a²

2a = k and a² = 9

a = 3, -3

If a = 3, then k = 2(3) ==> 6

If a = -3, then k = 2(-3) ==> -6

Problem 4 :

For all x ≠ 3, the expression above is equivalent to 

where k is positive constant. What is the value of k ?

Solution :

Problem 5 :

If  a = x² - 5x + 2 and b = 3x³ + 4x² - 6, what is 3a - b in terms of x ?

(a) 4x² - 15x + 12      (b)  -3x³ - 4x² - 15x + 12

(c) -3x³ - x² - 15x      (d) -3x³ + 7x² - 15x + 12

Solution :

a = x² - 5x + 2

3a = 3x² - 15x + 6

3a - b = 3x² - 15x + 6 - (3x³ + 4x² - 6)

= 3x² - 15x + 6 - 3x³ - 4x² + 6

= - 3x³ - x² - 15x + 12

Problem 6 :

Based on the equation above, which of the following is possible value of (x - 2) ?

(a)  √3     (b)  -2 + √3     (c)  2 - √3   (d) 3

Solution :

9 = 3(x² - 2x(2) + 2²)

9 = 3(x² - 4x + 2)

9 = 3x² - 12x + 6

3x² - 12x + 6 - 9 = 0

3x² - 12x - 3 = 0

Problem 7 :

If a⁴ - b² = 30 and a² + b = -5, what is the value of a² - b ?

(a)  -6   (b)  -2    (c) 3     (d)  6

Solution :

a⁴ - b² = 30

(a²)² - b² = 30

(a² + b)(a² - b) = 30

-5(a² - b) = 30

(a² - b) = 30/(-5)

(a² - b) = -6

Problem 8 :

If c = ab/d and d ≠ 0, then 1/ab = 

(a)  c + d   (b)  cd     (c) 1/cd    (d) c/d

Solution :

c = ab/d

c/d = ab

Taking reciprocal on both sides,

d/c = 1/ab

Problem 9 :

2x (x - y) (x + y) 

Which of the following is equivalent to the expression above ?

(a) 4x³ - 2xy²    (b)  2x³ + 2xy²

(c)  2x³ - 2xy²      (d)  2x³ - 4xy + 2xy²

Solution :

= 2x (x - y) (x + y) 

= 2x(x² - y²)

= 2x³ - 2xy²

Problem 10 :

If y⁸ = m and y⁹ = 2/3, what is the value of y in terms of m ?

(a)  2m/3   (b)  2/3m   (c) 3m/2   (D) 3/2m

Solution :

y⁸ = m -----(1)

y⁹ = 2/3 -----(2)

(2)/(1)  ==>  y⁹ / y⁸ = (2/3) / m

y = 2/3m

Problem 11 :

b(2a + 3) (2 + 5a)

What is the value of the coefficient of the ab term when the expression above is explained and the like terms are combined ?

Solution :

= b(2a + 3) (2 + 5a)

= b(4a + 10a² + 6 + 15a)

= b(10a² + 19a + 6)

Problem 12 :

If (x + 3) (x - 3) = 91, what is the value of x² ?

Solution :

(x + 3) (x - 3) = 91

x² - 3² = 91

x² - 9 = 91

Add 9 on both sides.

x² = 91 + 9

x² = 100

x = √100

x = ±10

Problem 13 :

a(3 - a) + 2(a + 5)

Which of the following is equivalent to the expression above ?

(a) -a² + 5a + 5     (b)  -a² + 5a + 10

(c)  4a + 5        (d)  4a + 10

Solution :

= a(3 - a) + 2(a + 5)

= 3a - a² + 2a + 10

= -a² + 2a + 3a + 10

= -a² + 5a + 10

So, option a is correct.

Problem 14 :

If y > 0 and 

(a)  -3/2   (b) -5/2   (c) 3/2   (d) 5/2

Solution :

Using properties of exponents, we get

The value of b is -3/2.