QUESTIONS ON OPERATIONS OF POLYNOMIAL FOR SAT

Problem 1 :

If 10xy - 3y + 6 = 41 + 2y, what is the value of 2xy - y ?

(a)  5   (b)  6   (c)  7   (d)  15

Solution :

10xy - 3y + 6 = 41 + 2y

Subtracting 2y and 6 on both sides, we get

10xy - 3y - 2y = 41 - 6

10xy - 5y = 35

Dividing by 5 on both sides.

2xy - y = 7

Problem 2 :

(a)  -3   (b)  -1/3     (c)  1/3    (d)  3

Solution :

Problem 3 :

If a = 2x³y² - 3x²y³  and b = -3x³y² + 2x²y³, what is a + b in terms of x and y ?

Solution :

a = 2x³y² - 3x²y³ ----(1)

b = -3x³y² + 2x²y³   ----(2)

a + b = 2x³y² - 3x²y³ -3x³y² + 2x²y³

a + b = -x³y² - x²y³

Problem 4 :

p(x) = (3x² - 5)(x + k) - 20

In the polynomial p(x) defined above, k is a constant. If x is a factor of p(x), what is the value of k ?

Solution :

Since x is a factor, x - 0 is also a factor.

x - 0 = 0

x = 0

p(x) = (3x² - 5)(x + k) - 20

p(0) = (3(0)² - 5)(0 + k) - 20

0 = -5k - 20

5k = -20

k = -20/5

k = -4

Problem 5 :

If (mx + c) (nx + 3) = 12x² + 5x - 3 for all values of x, where m, n and c are constants, what is the value of m + n ?

(a)  7   (b)  8  (c)  12   (d)  13

Solution :

(mx + c) (nx + 3) = 12x² + 5x - 3

mnx² + 3mx + cnx + 3c = 12x² + 5x - 3

mnx² + (3m + cn)x + 3c = 12x² + 5x - 3

Equating corresponding terms, we get

mn = 12 ----(1)

3m + cn = 5 ----(2)

3c = -3 ----(3)

c = -1

applying the value of c in (2), we get

3m - n = 5

From (1)

n = 12/m

3m - (12/m) = 5

3m²- 12 = 5m

3m²- 5m - 12 = 0

(m - 3)(3m + 4) = 0

m = 3 and m = -4/3

m + n = 3 + 4

m + n = 7

Problem 6 :

(75x² - 20) - 10(6 + 7x²)

The expression above can be written in the form a(x + b)(x - b), where a and b are positive constants. What is the value of a + b ?

Solution :

(75x² - 20) - 10(6 + 7x²)

= 75x² - 20 - 60 - 70x²

= 5x² - 80

= 5(x² - 16)

= 5(x² - 4²)

5 (x + 4)(x - 4) = a(x + b)(x - b)

a = 5 and b = 4

a + b = 4 + 5

a + b = 9

Problem 7 :

3(2xy + xyz + yz) - (3xy + 5xyz - 2yz)

It is equal to ?

Solution :

3(2xy + xyz + yz) - (3xy + 5xyz - 2yz)

Using distributive property, we get

= 6xy + 3xyz + 3yz - 3xy - 5xyz + 2yz

= 3xy - 2xyz + 5yz

Problem 8 :

Which of the following is equivalent to (m + n + 1) (m + n - 1) ?

(a) m² + 2mn + n² - 1    (b) m² - 2mn + n² - 1 

(c)  m² - n² - 1        (d)  m² + 2m + n² + 2n - 1 

Solution :

(m + n + 1) (m + n - 1)

Considering m + n as one term and 1 as another term. It looks like (a + b)(a - b).

= (m + n + 1) (m + n - 1)

= (m + n)² - 1²

= m²+ n² + 2mn - 1

= m² + 2mn + n² - 1

Problem 9 :

If a = 2/3b and ax = 5/6b for b ≠ 0, what is the value of x ?

Solution :

a = 2/3b ----(1)

ax = 5/6b ----(2)

Applying the value of a in (2), we get