Question

algebraic expressions
work 11: adding & subtracting polynomials

1. $-1 - 4a^2 + 17a$
2. $k - 17 - 2k^3 + 14k^2$

Explanation:

To solve the problem of arranging the polynomial in standard form, we need to order the terms from the highest degree to the lowest degree.

Step 1: Identify the degrees of each term

• For the term \(-4a^2\), the degree is \(2\) (since the exponent of \(a\) is \(2\)).
• For the term \(17a\), the degree is \(1\) (since the exponent of \(a\) is \(1\)).
• For the term \(-1\), the degree is \(0\) (since it is a constant term).

Step 2: Arrange the terms in descending order of their degrees

We arrange the terms from the highest degree to the lowest degree. So, the term with degree \(2\) (\(-4a^2\)) comes first, then the term with degree \(1\) (\(17a\)), and finally the constant term \(-1\).

So, the polynomial \(-1 - 4a^2 + 17a\) in standard form is:

$$ -4a^2 + 17a - 1 $$

For the second polynomial \(k - 17 - 2k^3 + 14k^2\)

Step 1: Identify the degrees of each term

• For the term \(-2k^3\), the degree is \(3\) (since the exponent of \(k\) is \(3\)).
• For the term \(14k^2\), the degree is \(2\) (since the exponent of \(k\) is \(2\)).
• For the term \(k\), the degree is \(1\) (since the exponent of \(k\) is \(1\)).
• For the term \(-17\), the degree is \(0\) (since it is a constant term).

Step 2: Arrange the terms in descending order of their degrees

We arrange the terms from the highest degree to the lowest degree. So, the term with degree \(3\) (\(-2k^3\)) comes first, then the term with degree \(2\) (\(14k^2\)), then the term with degree \(1\) (\(k\)), and finally the constant term \(-17\).

So, the polynomial \(k - 17 - 2k^3 + 14k^2\) in standard form is:

$$ -2k^3 + 14k^2 + k - 17 $$

Final Answers

1. \(\boldsymbol{-4a^2 + 17a - 1}\)
2. \(\boldsymbol{-2k^3 + 14k^2 + k - 17}\)

Answer:

To solve the problem of arranging the polynomial in standard form, we need to order the terms from the highest degree to the lowest degree.

Step 1: Identify the degrees of each term

• For the term \(-4a^2\), the degree is \(2\) (since the exponent of \(a\) is \(2\)).
• For the term \(17a\), the degree is \(1\) (since the exponent of \(a\) is \(1\)).
• For the term \(-1\), the degree is \(0\) (since it is a constant term).

Step 2: Arrange the terms in descending order of their degrees

We arrange the terms from the highest degree to the lowest degree. So, the term with degree \(2\) (\(-4a^2\)) comes first, then the term with degree \(1\) (\(17a\)), and finally the constant term \(-1\).

So, the polynomial \(-1 - 4a^2 + 17a\) in standard form is:

$$ -4a^2 + 17a - 1 $$

For the second polynomial \(k - 17 - 2k^3 + 14k^2\)

Step 1: Identify the degrees of each term

• For the term \(-2k^3\), the degree is \(3\) (since the exponent of \(k\) is \(3\)).
• For the term \(14k^2\), the degree is \(2\) (since the exponent of \(k\) is \(2\)).
• For the term \(k\), the degree is \(1\) (since the exponent of \(k\) is \(1\)).
• For the term \(-17\), the degree is \(0\) (since it is a constant term).

Step 2: Arrange the terms in descending order of their degrees

We arrange the terms from the highest degree to the lowest degree. So, the term with degree \(3\) (\(-2k^3\)) comes first, then the term with degree \(2\) (\(14k^2\)), then the term with degree \(1\) (\(k\)), and finally the constant term \(-17\).

So, the polynomial \(k - 17 - 2k^3 + 14k^2\) in standard form is:

$$ -2k^3 + 14k^2 + k - 17 $$

Final Answers

1. \(\boldsymbol{-4a^2 + 17a - 1}\)
2. \(\boldsymbol{-2k^3 + 14k^2 + k - 17}\)