Question 1

Suppose p(x) = x² + 6x + 4, q(x) = 6x³ - 4x + 1. Write the expression (p + q)(x) as a sum of terms, each of which is a constant times a power of x. A) 6x³ + x² + 2x + 5 B) 6x³ + x² + 2x - 5 C) 6x³ + x² - 2x + 5 D) 6x³ + x² - 2x - 5

Accepted Answer

Adding the polynomials p(x) and q(x) involves combining like terms. The unique terms from each polynomial (the terms with the superscript and subscript notation) are simply added together since they do not have like terms in the other polynomial. The linear and constant terms are combined by addition: (6x + (-4x)) = 2x and (4 + 1) = 5, resulting in the expression 6x³ + x² + 2x + 5.

Question 2

Suppose p(x) = x² + 9x + 7, r(x) = 18x³ + 9. Write the expression (9p - 7r)(x) as a sum of terms, each of which is a constant times a power of x.

Accepted Answer

The answer of Suppose p(x) = x² + 9x +...

Question 3

Suppose p(x) = x² + 3x + 1, r(x) = 6x³ + 3. Write the expression (p(x))²(r(x) - 3) as a sum of terms, each of which is a constant times a power of x.

Accepted Answer

The answer of Suppose p(x) = x² + 3x +...

Question 4

Suppose q(x) = 5x³ - 3x + 1, r(x) = 10x³ + 5. Write the expression as a sum of terms, each of which is a constant times a power of x.

Accepted Answer

The answer of Suppose q(x) = 5x³ - 3x +...

Question 5

Suppose p(x) = x² + 4x + 2, q(x) = 4x³ - 2x + 1, r(x) = 8x³ + 4. Write the expression as a sum of terms, each of which is a constant times a power of x.

Accepted Answer

The answer of Suppose p(x) = x² + 4x +...

Question 6

Suppose p(x) = x² + 8x + 6. Write the expression As a sum of terms, each of which is a constant times a power of h. A) 10 - h B) 10 + h C) -6 + h D) -6 - h

Accepted Answer

The answer of Suppose p(x) = x² + 8x +...

Question 7

Find all real numbers x such that x⁴ + 36x² - 2,592 = 0

Accepted Answer

The answer of Find all real numbers x such that x⁴...

Question 8

Find all real numbers x such that: 
A) 4, 1
B) -4, 1
C) 4, -1
D) -4, -1

Accepted Answer

The answer of Find all real numbers x such that:...

Question 9

Find all real numbers x such that: x⁴ - 53x² + 196 = 0

Accepted Answer

The answer of Find all real numbers x such that: x⁴...

Question 10

Suppose q(x) = $x^3 - 49x + b$. The point (7, b) lies on the graph of q.

Accepted Answer

Substituting x = 7 into the equation q(x) = $x^3 - 49x + b$ gives $7^3 - 49*7 + b = b$, which means the point (7, b) satisfies the equation and thus lies on the graph of q.

Question 11

is $x^8 - 1 = (x^4 + 1)(x^2 + 1)(x + 1)(x - 1)$ correct?

Accepted Answer

The equation is correct because it represents the factorization of $x^8 - 1$ using the difference of squares and sum of squares identities.

Question 12

Find a number b such that 6 is a zero of the polynomial p defined by p(x) = -36 + bx - 6x² + x³. A) 36 B) -36 C) 6 D) -6

Accepted Answer

The answer of Find a number b such that 6...

Question 13

Find a number b such that 1 is a zero of the polynomial p defined by p(x) = b - 9x + bx² + 9x³.

Accepted Answer

The answer of Find a number b such that 1...

Question 14

Find a polynomial p of degree 3 such that -6, 6, and 7 are zeros of p and p(0) = 36. Write your answer in descending order by degree.

Accepted Answer

The answer of Find a polynomial p of degree 3...

Question 15

Find a polynomial p of degree 3 such that -1, 
  , and 1 are zeros of p and p(0) =   . Write your answer in descending order by degree.

Accepted Answer

The answer of Find a polynomial p of degree 3...

Question 16

If p(x) = a²x² + 4x + 5 and q(x) = (1 - 2a)x² - 8x - 5, find a real number a such that (p + q)(x) has degree 1. A) 4 B) -4 C) 0 D) 1

Accepted Answer

The answer of If p(x) = a²x² + 4x +...

Question 17

Suppose d is a real number. Then $(d + 6)^4 = d^4 + 6^4$ if and only if d = 0

Accepted Answer

$(d + 6)^4 = d^4 + 6^4$ is a binomial expansion. The only way for a binomial expansion to be equal to the sum of the squares of the two terms is if one of the terms is 0. Therefore, d = 0.

Question 18

The only possible integer zeros of $p(x) = 2x^5 - bx^3 + 4x - 2$ are 1 and 2.

Accepted Answer

The Rational Root Theorem suggests that the possible rational zeros of a polynomial are given by the factors of the constant term divided by the factors of the leading coefficient. For $p(x) = 2x^5 - bx^3 + 4x - 2$, the constant term is -2 and the leading coefficient is 2. Therefore, the possible integer zeros are ±1, ±2, ±(2/2), which simplifies to ±1, ±2. This means -1 is also a possible integer zero, not just 1 and 2.

Question 19

Find all choices of (b, c, d) such that 4 and 1 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Accepted Answer

The answer of Find all choices of (b, c, d)...

Question 20

Find all choices of (b, c, d) such that 9 and -9 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Accepted Answer

The answer of Find all choices of (b, c, d)...

Quiz 3: Linear, Quadratic, Polynomial, and Rational Functions

Suppose p(x) = x² + 6x + 4, q(x) = 6x³ - 4x + 1. Write the expression (p + q) (x) as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 9x + 7, r(x) = 18x³ + 9. Write the expression (9p - 7r)(x) as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 3x + 1, r(x) = 6x³ + 3. Write the expression (p(x))²(r(x) - 3) as a sum of terms, each of which is a constant times a power of x.

Suppose q(x) = 5x³ - 3x + 1, r(x) = 10x³ + 5. Write the expression as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 4x + 2, q(x) = 4x³ - 2x + 1, r(x) = 8x³ + 4. Write the expression as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 8x + 6. Write the expression As a sum of terms, each of which is a constant times a power of h.

Find all real numbers x such that x⁴ + 36x² - 2,592 = 0

Find all real numbers x such that:

Find all real numbers x such that: x⁴ - 53x² + 196 = 0

Suppose q(x) = $x^3 - 49x + b$ . The point (7, b) lies on the graph of q.

is $x^8 - 1 = (x^4 + 1)(x^2 + 1)(x + 1)(x - 1)$ correct?

Find a number b such that 6 is a zero of the polynomial p defined by p(x) = -36 + bx - 6x² + x³.

Find a number b such that 1 is a zero of the polynomial p defined by p(x) = b - 9x + bx² + 9x³.

Find a polynomial p of degree 3 such that -6, 6, and 7 are zeros of p and p(0) = 36. Write your answer in descending order by degree.

Find a polynomial p of degree 3 such that -1, , and 1 are zeros of p and p(0) = . Write your answer in descending order by degree.

If p(x) = a²x² + 4x + 5 and q(x) = (1 - 2a) x² - 8x - 5, find a real number a such that (p + q) (x) has degree 1.

Suppose d is a real number. Then $(d + 6)^4 = d^4 + 6^4$ if and only if d = 0

The only possible integer zeros of $p(x) = 2x^5 - bx^3 + 4x - 2$ are 1 and 2.

Find all choices of (b, c, d) such that 4 and 1 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Find all choices of (b, c, d) such that 9 and -9 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Quiz 3: Linear, Quadratic, Polynomial, and Rational Functions

Suppose p(x) = x2 + 6x + 4, q(x) = 6x3 - 4x + 1. Write the expression (p + q) (x) as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x2 + 9x + 7, r(x) = 18x3 + 9. Write the expression (9p - 7r)(x) as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x2 + 3x + 1, r(x) = 6x3 + 3. Write the expression (p(x))2(r(x) - 3) as a sum of terms, each of which is a constant times a power of x.

Suppose q(x) = 5x3 - 3x + 1, r(x) = 10x3 + 5. Write the expression as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x2 + 4x + 2, q(x) = 4x3 - 2x + 1, r(x) = 8x3 + 4. Write the expression as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x2 + 8x + 6. Write the expression As a sum of terms, each of which is a constant times a power of h.

Find all real numbers x such that x4 + 36x2 - 2,592 = 0

Find all real numbers x such that:

Find all real numbers x such that: x4 - 53x2 + 196 = 0

Suppose q(x) = x3−49x+bx^3 - 49x + bx3−49x+b . The point (7, b) lies on the graph of q.

is x8−1=(x4+1)(x2+1)(x+1)(x−1)x^8 - 1 = (x^4 + 1)(x^2 + 1)(x + 1)(x - 1)x8−1=(x4+1)(x2+1)(x+1)(x−1) correct?

Find a number b such that 6 is a zero of the polynomial p defined by p(x) = -36 + bx - 6x2 + x3.

Find a number b such that 1 is a zero of the polynomial p defined by p(x) = b - 9x + bx2 + 9x3.

Find a polynomial p of degree 3 such that -6, 6, and 7 are zeros of p and p(0) = 36. Write your answer in descending order by degree.

Find a polynomial p of degree 3 such that -1, , and 1 are zeros of p and p(0) = . Write your answer in descending order by degree.

If p(x) = a2x2 + 4x + 5 and q(x) = (1 - 2a) x2 - 8x - 5, find a real number a such that (p + q) (x) has degree 1.

Suppose d is a real number. Then (d+6)4=d4+64(d + 6)^4 = d^4 + 6^4(d+6)4=d4+64 if and only if d = 0

The only possible integer zeros of p(x)=2x5−bx3+4x−2p(x) = 2x^5 - bx^3 + 4x - 2p(x)=2x5−bx3+4x−2 are 1 and 2.

Find all choices of (b, c, d) such that 4 and 1 are the only zeros of the polynomial p(x) = x3 + bx2 + cx + d.

Find all choices of (b, c, d) such that 9 and -9 are the only zeros of the polynomial p(x) = x3 + bx2 + cx + d.

Suppose p(x) = x² + 6x + 4, q(x) = 6x³ - 4x + 1. Write the expression (p + q) (x) as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 9x + 7, r(x) = 18x³ + 9. Write the expression (9p - 7r)(x) as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 3x + 1, r(x) = 6x³ + 3. Write the expression (p(x))²(r(x) - 3) as a sum of terms, each of which is a constant times a power of x.

Suppose q(x) = 5x³ - 3x + 1, r(x) = 10x³ + 5. Write the expression as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 4x + 2, q(x) = 4x³ - 2x + 1, r(x) = 8x³ + 4. Write the expression as a sum of terms, each of which is a constant times a power of x.

Suppose p(x) = x² + 8x + 6. Write the expression As a sum of terms, each of which is a constant times a power of h.

Find all real numbers x such that x⁴ + 36x² - 2,592 = 0

Find all real numbers x such that: x⁴ - 53x² + 196 = 0

Suppose q(x) = $x^3 - 49x + b$ . The point (7, b) lies on the graph of q.

is $x^8 - 1 = (x^4 + 1)(x^2 + 1)(x + 1)(x - 1)$ correct?

Find a number b such that 6 is a zero of the polynomial p defined by p(x) = -36 + bx - 6x² + x³.

Find a number b such that 1 is a zero of the polynomial p defined by p(x) = b - 9x + bx² + 9x³.

If p(x) = a²x² + 4x + 5 and q(x) = (1 - 2a) x² - 8x - 5, find a real number a such that (p + q) (x) has degree 1.

Suppose d is a real number. Then $(d + 6)^4 = d^4 + 6^4$ if and only if d = 0

The only possible integer zeros of $p(x) = 2x^5 - bx^3 + 4x - 2$ are 1 and 2.

Find all choices of (b, c, d) such that 4 and 1 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.

Find all choices of (b, c, d) such that 9 and -9 are the only zeros of the polynomial p(x) = x³ + bx² + cx + d.