MACHING COEFFICIENTS OF POLYNOMIALS PRACTICE FOR SAT

Problem 1 :

(x - c)² = x + 3

If c = 3, what is the solution set of the equation above ?

a)  {1}      b)  {6}       c) {1, 6}      d)  {-3, 1, 6}

Solution :

(x - c)² = x + 3

Applying c = 3,

(x - 3)² = x + 3

x² - 2x(3) + 3² = x + 3

x² - 6x + 9 = x + 3

x² - 6x - x + 9 - 3 = 0

x² - 7x + 6 = 0

(x - 1) (x - 6) = 0

Equating each factor to 0, we get

x = 1 and x = 6

Problem 2 :

5x + 12 = (10x + 3c) / 2

In the equation above, c is a constant. For what value of c will the equation have infinitely many solutions ?

Solution :

5x + 12 = (10x + 3c) / 2

2(5x + 12) = 10x + 3c

10x + 24 = 10x + 3c

Here the given condition is to have infinitely many solution. All the values of x which is satisfying the above equation can be considered as solution.

By getting 24 as constant in the right side, we can get same values on both side of the equal sign.

3c = 24, then c = 8

Problem 3 :

In the xy plane, the points (c, 2d) and (c + 3, 4d) lie on the line with equation y = mx + b, where m and b are nonzero constants/ What is the value of d/m ?

a)  2/3    b)  1    c)  3/2    d)  2

Solution :

Slope (m) = (y₂ - y₁) / (x₂ - x₁) 

= (4d - 2d) / (c + 3 - c) 

m = 2d / 3

Doing cross multiplication, we get

3m = 2d

d/m = 3/2

So, the answer is option c.

Problem 4 :

If the expression (1/4) x² + 3x + 9 is rewritten in the form 1/4 (x + a)², where a is a positive constant. What is the value of a ?

a)   3/2    b)  3     c)   6       d) 2√3

Solution :

(1/4) x² + 3x + 9

= (1/4)[x² + 12x + 36]

= (1/4)[x² + 2 x (6) + 6²]

= (1/4)(x + 6)² 

Comparing with 1/4 (x + a)²

x = 6

So, the answer is option c.

Problem 5 :

In the xy-plane, the line defined by the equation y = 3x - 5 passes through the vertex of a parabola with x-intercepts 3 and 15. What is the y-coordinate of the vertex of the parabola ?

Solution :

x-coordinate of the vertex will be the middle of the x-intercepts.

x = 3 and x = 15

midpoint = (3 + 15) / 2

= 18/2

= 9

x-coordinate of vertex is 9. Let y be the unknown coordinate of vertex.

When x = 9, then y = 3x - 5

y = 3(9) - 5

y = 27 - 5

y = 22

Then y-coordinate is 22.

Problem 6 :

9x³ - kx + 4

In the polynomial above, k is an integer. If 3x - 2 is a factor of the polynomial. What is the value of k ?

Solution :

9x³ - kx + 4

Since 3x - 2 is factor, 3x - 2 = 0

x = 2/3 is a solution.

Let f(x) = 9x³ - kx + 4

f(2/3) = 9(2/3)³ - k(2/3) + 4

0 = 9(8/27) - k(2/3) + 4

8/3 - 2k/3 + 4 = 0 

(8 - 2k + 12) / 3 = 0

20 - 2k = 0

k = 10

So, the value of k is 10.

Problem 7 :

The function f a nd g are defined by f(x) = x² + 2 and g(x) = 4x  - 3. If a > 0, for what value of a does g(f(a)) = 41 ?

Solution :

f(x) = x² + 2 and g(x) = 4x  - 3

f(a) = a² + 2

g(f(a)) = 4(a² + 2) - 3

Given that g(f(a)) = 41

4(a² + 2) - 3 = 41

4(a² + 2) = 41 + 3

4(a² + 2) = 44

(a² + 2) = 11

a² = 11 - 2

a² = 9

a = 3 and -3

So, the value of a is -3 and 3.

Problem 8 :

In the xy-plane the line with equation y = ax + b, where a and b are constants, intersects the line with equation y = 2bx + a at the point (3, 4) .If b ≠ 0, what is the value of a/b ?

a) 2/3   b) 3/4   c)  5/2    d) 7/3

Solution :

The point (3, 4) lies on the line y = ax + b

4 = a(3) + b

4 = 3a + b -----(1)

The point (3, 4) lies on the line y = 2bx + a

4 = 2b(3) + a

4 = 6b + a -----(2)

(1) - (2)

2a - 5b = 0

2a = 5b

a/b = 5/2

So, option c is correct.

Problem 9 :

y = x² - k

In the equation above, k is a constant. If the graph of the equation in the xy-plane is a parabola with x-intercepts of -4 and 4, what is the minimum value of y in terms of k ?

Solution :

y = x² - k

The minimum will appear the midpoint of x-intercepts.

(-4 + 4)/2

= 0

The x-coordinate of minimum is x = 0

y = 0² - k

y = - k

Problem 10 :

In the xy - plane the points (a, 7) and (b, 12) lie on the graph of y = x² + 3. What is the minimum possible value of a + b ?

a)  -5       b)  -1    c)   1    d)  5

Solution :

The point (a, 7) lies on the parabola y = x² + 3

The point (b, 12) lies on the parabola y = x² + 3

Least value of a = -2 and b = -3.

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

Minimum value of a + b = -5