1. A polynomial is a finite series of terms in the form ax^m where is a(n)------ and m is a(n)---------
Real number
Irrational number
whole number
rational number
2. Classify each expression below as either a polynomial or not.
Polynomial
Not a Polynomial
x-3+ \sqrt{ }x
-4x^8
X^4+2y^2+z-w^2
X^-3-x+1
3. Explain why 2x^3y and 2yx^2 are not like terms
4. Which two polynomials have a sum of 4x−6?
A. 4x^2+x−3) +(4x^2+3x+3)
B. (−4x^2+x−3) +(4x^2−5x−3)
C. (−4x^2+x−3) +(4x^2+3x−3)
D. (−4x^2+x−3) +(−4x^2+3x+3)
5. Subtract (2x^2+5x−3) −(x^2−2x+4)
A. 3x^2+3x+1
B. X^2+3x+1
C. X^2+7x-1
D. X^2+7x-7
6. Multiply (x−4)(x−5)
A. X^2-9x+20
B. X^2+9x-20
C. x2−9x-20
D. x^2+9x+20
7. Multiply (x-4)^2
A. X^2+16
B. X^2-8x+16
C. X^2+8x-16
D. X^2-8x-16
8. Given the expression)(−2x^2)(3x)(x) Jose says that you will get different results if you change the order that you multiply the quantities. Is Jose correct? Explain.
A. No because multiplication is commutative.
B. Yes, because multiplication is not associative.
C. Yes, because multiplication is not commutative.
D. No becuase multiplication is associative.
9. Completely factor the following expression ab^3+2a^2b^2
A. ab(b^2+2ab)
B. b^2(ab+2a^2
C. a(b^3+2ab^2)
D. ab^2(b+2a)
10. Completely factor the following expression: x^2+x−6
A. (x+6)(x−1)
B. (x−3)(x+2)
C. (x−3)(x-2)
D. (x−6)(x+1)
11. Completely factor the following expression x^2−7x+12
A. (x−4)(x−3)
B. (x+4)(x-3)
C. (x−4)(x+4
D. (x+4)(x+3)
12. Factor the following expression completely: X^2+6x+9
A. (x−3)(x+3)
B. (x+3)^2
C. (x+6)(x+9)
D. (x-3)^2
13. Factor the following expression completely 4x^2+5x−21
A. (4x−3)(x+7)
B. (2x+3)(2x−7)
C. (4x−7)(x+3)
D. (2x+7)(2x−3)
14. Factor the following expression completely 3x^2-75
A. (3x+15)(x−5)
B. (3x−15)(x+5)
C. 3(x−5)(x+5)
D. 3(x−5)(x−5)
15. Factor the following expression completely: 9x^2−16
A. 9(x−4)(x+4)
B. (3x−4)(3x+4)
C. (3x−4)(3x−4)
D. (3x+4)(3x+4)
please answer all of them
1. A polynomial is a finite series of terms in the form ax^m where a is a real number and m is a whole number.
2. Classify each expression below as either a polynomial or not.
a) x-3+ \sqrt{ }x - Not a Polynomial (due to the square root term)
b) -4x^8 - Polynomial
c) X^4+2y^2+z-w^2 - Polynomial
d) X^-3-x+1 - Not a Polynomial (due to negative exponent)
3. 2x^3y and 2yx^2 are not like terms because the exponent of x in 2x^3y is 3, whereas the exponent of x in 2yx^2 is 2. Like terms must have the same variables raised to the same power.
4. The two polynomials that have a sum of 4x−6 are:
B. (−4x^2+x−3) +(4x^2−5x−3)
5. Subtracting (2x^2+5x−3) −(x^2−2x+4) gives
C. x^2+7x-1
6. Multiplying (x−4)(x−5) gives
A. x^2-9x+20
7. Multiplying (x-4)^2 gives
B. x^2-8x+16
8. Jose is correct. The order of multiplying the quantities does matter. Multiplication is not commutative.
9. Completely factoring the expression ab^3+2a^2b^2 gives
C. a(b^3+2ab^2)
10. Completely factoring the expression x^2+x−6 gives
B. (x−3)(x+2)
11. Completely factoring the expression x^2−7x+12 gives
A. (x−4)(x−3)
12. Factoring the expression completely: X^2+6x+9 gives
B. (x+3)^2
13. Factoring the expression completely 4x^2+5x−21 gives
A. (4x−3)(x+7)
14. Factoring the expression completely 3x^2-75 gives
C. 3(x−5)(x+5)
15. Factoring the expression completely: 9x^2−16 gives
A. 9(x−4)(x+4)
1. isaac was asked to factor the following expression completely: 8x^2−20x−12 Isaac got (8x+4) (x−3) as their answer and claims it is correct because when multiplying the factors it simplifies to the original polynomial expression. Is Isaac correct? Why or why not?
2. Classify each of the following polynomials as factorable or not factorable.
Factorable
Not Factorable
X^2−2x−15
X^2+5x+2
X^2+6x+8
X^2+-5x+3
3. Explain why we know that 3x^2+4x−2 is not factorable.
1. Isaac is incorrect. The correct factored form of 8x^2−20x−12 is (2x-6)(4x+2), not (8x+4)(x−3). When multiplying the factors (2x-6)(4x+2), it simplifies to the original polynomial expression.
2. Classify each of the following polynomials as factorable or not factorable.
a) X^2−2x−15 - Factorable
b) X^2+5x+2 - Factorable
c) X^2+6x+8 - Factorable
d) X^2+-5x+3 - Not factorable
3. We know that 3x^2+4x−2 is not factorable because it cannot be written as the product of two binomials with integer coefficients. It does not have any integer factors that can be multiplied to obtain the original polynomial.
1. a) Real number, b) whole number.
2. a) Not a Polynomial, b) Polynomial, c) Polynomial, d) Polynomial.
3. 2x^3y and 2yx^2 are not like terms because the exponent of x is different in each term. In 2x^3y, the exponent of x is 3, while in 2yx^2, the exponent of x is 2. For two terms to be like terms, they must have the same variables raised to the same power.
4. The correct answer is B. (-4x^2+x-3) +(4x^2-5x-3)
5. The correct answer is D. x^2+7x-7
6. The correct answer is A. x^2-9x+20
7. The correct answer is B. x^2-8x+16
8. A. No because multiplication is commutative. Changing the order of multiplication does not change the result.
9. The correct answer is C. a(b^3+2ab^2)
10. The correct answer is B. (x-3)(x+2)
11. The correct answer is A. (x-4)(x-3)
12. The correct answer is B. (x+3)^2
13. The correct answer is A. (4x-3)(x+7)
14. The correct answer is C. 3(x-5)(x+5)
15. The correct answer is A. 9(x-4)(x+4)
Sure! Here are the answers to all the questions:
1. A polynomial is a finite series of terms in the form ax^m where "a" is a Real number and "m" is a whole number.
2. Classify each expression below as either a polynomial or not.
- x-3+sqrt(x) is not a polynomial because it contains a square root, which is not in the form of a polynomial term.
- -4x^8 is a polynomial because it is in the form of a polynomial term.
- x^4+2y^2+z-w^2 is a polynomial because all the terms are in the form of a polynomial term.
- x^-3-x+1 is not a polynomial because it contains a negative exponent, which is not in the form of a polynomial term.
3. 2x^3y and 2yx^2 are not like terms because the variables and their exponents are ordered differently. In like terms, the variables should be the same and their exponents should be the same as well. In this case, 2x^3y has the variable x raised to the power of 3 and y raised to the power of 1, while 2yx^2 has the variable x raised to the power of 2 and y raised to the power of 1.
4. The polynomial that has a sum of 4x-6 can be found by adding the two given polynomials:
- (4x^2+x-3) + (4x^2+3x+3) = 8x^2 + 4x - 6
5. To subtract (2x^2+5x-3) from (x^2-2x+4), we need to distribute the negative sign to every term inside the parentheses, and then combine like terms:
- (x^2-2x+4) - (2x^2+5x-3) = x^2 - 2x + 4 - 2x^2 - 5x + 3 = -x^2 - 7x + 7
Therefore, the answer is C. x^2 + 7x - 1.
6. To multiply (x-4)(x-5), we need to use the FOIL method:
- (x-4)(x-5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20
Therefore, the answer is A. x^2 - 9x + 20.
7. To multiply (x-4)^2, we can use the formula for squaring a binomial:
- (x-4)^2 = x^2 - 8x + 16
Therefore, the answer is B. x^2 - 8x + 16.
8. Jose is correct. If we change the order of multiplication for (-2x^2)(3x)(x), we would get different results. Multiplication is not commutative, which means that changing the order of the factors will result in a different product. Therefore, the correct answer is C. Yes, because multiplication is not commutative.
9. To completely factor the expression ab^3 + 2a^2b^2, we can factor out the common terms:
- ab^3 + 2a^2b^2 = ab^2(b + 2a)
Therefore, the answer is A. ab^2(b + 2a).
10. To completely factor the expression x^2 + x - 6, we can use factoring methods:
- (x−3)(x+2)
Therefore, the answer is B. (x−3)(x+2).
11. To completely factor the expression x^2 - 7x + 12, we can use factoring methods:
- (x−4)(x−3)
Therefore, the answer is A. (x−4)(x−3).
12. To factor the expression x^2 + 6x + 9 completely, we recognize that it is a perfect square trinomial and can be factored as:
- (x+3)^2
Therefore, the answer is B. (x+3)^2.
13. To factor the expression 4x^2 + 5x - 21 completely, we can use factoring methods:
- (4x−3)(x+7)
Therefore, the answer is A. (4x−3)(x+7).
14. To factor the expression 3x^2 - 75 completely, we can factor out the greatest common factor first:
- 3(x^2 - 25)
Then we can factor the difference of squares within the parentheses:
- 3(x-5)(x+5)
Therefore, the answer is C. 3(x-5)(x+5).
15. To factor the expression 9x^2 - 16 completely, we recognize that it is a difference of squares and can be factored as:
- (3x-4)(3x+4)
Therefore, the answer is B. (3x-4)(3x+4).