Solved on Dec 11, 2023

Write a degree 2 trinomial with leading coef 8 and constant -5. Write two degree 1 binomials, then multiply and simplify.
$8 x^{2} - 5$

STEP 1

Assumptions
1. A trinomial of degree 2 is a polynomial with three terms, the highest degree term being $x^2$ .
2. The leading coefficient is the coefficient of the highest degree term.
3. The constant term is the term without a variable.
4. A binomial of degree 1 is a polynomial with two terms, the highest degree term being $x$ .
5. The constant term in a polynomial is the term that does not contain any variable.
6. Multiplying two binomials involves using the distributive property (FOIL method).

STEP 2

To write a trinomial of degree 2 with a leading coefficient of 8 and a constant term of -5, we need to find a middle term that will make the polynomial a trinomial. The general form of a trinomial of degree 2 is:
$ax^2 + bx + c$
where $a$ is the leading coefficient, $b$ is the coefficient of the $x$ term, and $c$ is the constant term.

STEP 3

Since we are given $a = 8$ and $c = -5$ , we can choose any value for $b$ to create the trinomial. Let's choose $b = 0$ for simplicity.
$8x^2 + 0x - 5$

STEP 4

We can simplify the expression by removing the term with a coefficient of 0:
$8x^2 - 5$
This is a trinomial of degree 2 with a leading coefficient of 8 and a constant term of -5.

STEP 5

To write a binomial of degree 1 with a leading coefficient of 2, we use the general form of a binomial of degree 1:
$ax + b$
where $a$ is the leading coefficient and $b$ is the constant term.

STEP 6

Since we are given $a = 2$ , we can choose any value for $b$ . Let's choose $b = 0$ for simplicity.
$2x + 0$

STEP 7

We can simplify the expression by removing the term with a coefficient of 0:
$2x$
This is a binomial of degree 1 with a leading coefficient of 2.

STEP 8

To write a binomial of degree 1 with a constant term of -3, we use the same general form of a binomial of degree 1:
$ax + b$

STEP 9

Since we are given $b = -3$ , we can choose any value for $a$ . Let's choose $a = 1$ for simplicity.
$1x - 3$

STEP 10

We can simplify the expression by removing the coefficient of 1:
$x - 3$
This is a binomial of degree 1 with a constant term of -3.

STEP 11

Now we need to multiply the two binomials we created in steps b) and c):
$2x \cdot (x - 3)$

STEP 12

We use the distributive property to multiply each term in the first binomial by each term in the second binomial:
$2x \cdot x + 2x \cdot (-3)$

STEP 13

Multiply the terms:
$2x^2 - 6x$

STEP 14

The expression $2x^2 - 6x$ is already in standard form, which is a polynomial arranged in descending order of degree.

STEP 15

Draw a rectangle around the final answer:
$\boxed{2x^2 - 6x}$

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Write a degree 2 trinomial with leading coef 8 and constant -5. Write two degree 1 binomials, then multiply and simplify.
$8 x^{2} - 5$

STEP 1

STEP 2

STEP 3

Since we are given $a = 8$ and $c = -5$ , we can choose any value for $b$ to create the trinomial. Let's choose $b = 0$ for simplicity.
$8x^2 + 0x - 5$

STEP 4

We can simplify the expression by removing the term with a coefficient of 0:
$8x^2 - 5$
This is a trinomial of degree 2 with a leading coefficient of 8 and a constant term of -5.

STEP 5

To write a binomial of degree 1 with a leading coefficient of 2, we use the general form of a binomial of degree 1:
$ax + b$
where $a$ is the leading coefficient and $b$ is the constant term.

STEP 6

Since we are given $a = 2$ , we can choose any value for $b$ . Let's choose $b = 0$ for simplicity.
$2x + 0$

STEP 7

We can simplify the expression by removing the term with a coefficient of 0:
$2x$
This is a binomial of degree 1 with a leading coefficient of 2.

STEP 8

To write a binomial of degree 1 with a constant term of -3, we use the same general form of a binomial of degree 1:
$ax + b$

STEP 9

Since we are given $b = -3$ , we can choose any value for $a$ . Let's choose $a = 1$ for simplicity.
$1x - 3$

STEP 10

We can simplify the expression by removing the coefficient of 1:
$x - 3$
This is a binomial of degree 1 with a constant term of -3.

STEP 11

Now we need to multiply the two binomials we created in steps b) and c):
$2x \cdot (x - 3)$

STEP 12

We use the distributive property to multiply each term in the first binomial by each term in the second binomial:
$2x \cdot x + 2x \cdot (-3)$

STEP 13

Multiply the terms:
$2x^2 - 6x$

STEP 14

The expression $2x^2 - 6x$ is already in standard form, which is a polynomial arranged in descending order of degree.

STEP 15

Draw a rectangle around the final answer:
$\boxed{2x^2 - 6x}$

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Write a degree 2 trinomial with leading coef 8 and constant -5. Write two degree 1 binomials, then multiply and simplify.8x2−58 x^{2} - 58x2−5

STEP 1

STEP 2

STEP 3

Since we are given a=8a = 8a=8 and c=−5c = -5c=−5, we can choose any value for bbb to create the trinomial. Let's choose b=0b = 0b=0 for simplicity.8x2+0x−58x^2 + 0x - 58x2+0x−5

STEP 4

We can simplify the expression by removing the term with a coefficient of 0:8x2−58x^2 - 58x2−5This is a trinomial of degree 2 with a leading coefficient of 8 and a constant term of -5.

STEP 5

To write a binomial of degree 1 with a leading coefficient of 2, we use the general form of a binomial of degree 1:ax+bax + bax+bwhere aaa is the leading coefficient and bbb is the constant term.

STEP 6

Since we are given a=2a = 2a=2, we can choose any value for bbb. Let's choose b=0b = 0b=0 for simplicity.2x+02x + 02x+0

STEP 7

We can simplify the expression by removing the term with a coefficient of 0:2x2x2xThis is a binomial of degree 1 with a leading coefficient of 2.

STEP 8

To write a binomial of degree 1 with a constant term of -3, we use the same general form of a binomial of degree 1:ax+bax + bax+b

STEP 9

Since we are given b=−3b = -3b=−3, we can choose any value for aaa. Let's choose a=1a = 1a=1 for simplicity.1x−31x - 31x−3

STEP 10

We can simplify the expression by removing the coefficient of 1:x−3x - 3x−3This is a binomial of degree 1 with a constant term of -3.

STEP 11

Now we need to multiply the two binomials we created in steps b) and c):2x⋅(x−3)2x \cdot (x - 3)2x⋅(x−3)

STEP 12

We use the distributive property to multiply each term in the first binomial by each term in the second binomial:2x⋅x+2x⋅(−3)2x \cdot x + 2x \cdot (-3)2x⋅x+2x⋅(−3)

STEP 13

Multiply the terms:2x2−6x2x^2 - 6x2x2−6x

STEP 14

The expression 2x2−6x2x^2 - 6x2x2−6x is already in standard form, which is a polynomial arranged in descending order of degree.

STEP 15

Draw a rectangle around the final answer:2x2−6x\boxed{2x^2 - 6x}2x2−6x​

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Write a degree 2 trinomial with leading coef 8 and constant -5. Write two degree 1 binomials, then multiply and simplify.
$8 x^{2} - 5$

Since we are given $a = 8$ and $c = -5$ , we can choose any value for $b$ to create the trinomial. Let's choose $b = 0$ for simplicity.
$8x^2 + 0x - 5$

We can simplify the expression by removing the term with a coefficient of 0:
$8x^2 - 5$
This is a trinomial of degree 2 with a leading coefficient of 8 and a constant term of -5.

To write a binomial of degree 1 with a leading coefficient of 2, we use the general form of a binomial of degree 1:
$ax + b$
where $a$ is the leading coefficient and $b$ is the constant term.

Since we are given $a = 2$ , we can choose any value for $b$ . Let's choose $b = 0$ for simplicity.
$2x + 0$

We can simplify the expression by removing the term with a coefficient of 0:
$2x$
This is a binomial of degree 1 with a leading coefficient of 2.

To write a binomial of degree 1 with a constant term of -3, we use the same general form of a binomial of degree 1:
$ax + b$

Since we are given $b = -3$ , we can choose any value for $a$ . Let's choose $a = 1$ for simplicity.
$1x - 3$

We can simplify the expression by removing the coefficient of 1:
$x - 3$
This is a binomial of degree 1 with a constant term of -3.

Now we need to multiply the two binomials we created in steps b) and c):
$2x \cdot (x - 3)$

We use the distributive property to multiply each term in the first binomial by each term in the second binomial:
$2x \cdot x + 2x \cdot (-3)$

Multiply the terms:
$2x^2 - 6x$

The expression $2x^2 - 6x$ is already in standard form, which is a polynomial arranged in descending order of degree.

Draw a rectangle around the final answer:
$\boxed{2x^2 - 6x}$