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Simplify the following algebraic expressions: 2.1.1 -2x - x3 + 9x2 + 1 2.2.1 -8 - 9h + 2h + 13 2.2.2 3ab + 2a + b + a - 3b2 - 3a - 4ba + 5b2 - b

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Part 2

Part 2.1· 3 steps

Given the expression −2x−x3+9x2+1-2x - x^3 + 9x^2 + 1.

  • 2.1.1 How many terms does the expression consist of? The terms are −2x-2x, −x3-x^3, 9x29x^2, and 1. There are 4 terms. 4\boxed{\text{4}}

  • 2.1.2 What is the coefficient of xx? The term containing xx is −2x-2x. The coefficient of xx is −2-2. −2\boxed{-2}

  • 2.1.3 What is the constant term? The constant term is the term without any variables. The constant term is 1. 1\boxed{1}

  • 2.1.4 Determine the value of the expression if x=−2x = -2.

  1. Substitute x=−2x = -2 into the expression.

    $$
-2(-2) - (-2)^3 + 9(-2)^2 + 1
$$

  2. Calculate the powers and products.

    $$
4 - (-8) + 9(4) + 1
$$
$$
4 + 8 + 36 + 1
$$

  3. Add the terms.

    $$
12 + 36 + 1 = 49
$$
$\boxed{49}$

Part 2.2· 5 steps

Simplify

  • 2.2.1 −8−9h+2h+13-8 - 9h + 2h + 13

  1. Group like terms.

    $$
(-8 + 13) + (-9h + 2h)
$$

  2. Combine the terms.

    $$
5 - 7h
$$
$\boxed{5 - 7h}$
    • 2.2.2 3ab+2a+b+a−3b2−3a−4ba+5b2−b3ab + 2a + b + a - 3b^2 - 3a - 4ba + 5b^2 - b

  3. Identify and group like terms (baba is the same as abab).

    $$
(3ab - 4ab) + (2a + a - 3a) + (b - b) + (-3b^2 + 5b^2)
$$

  4. Combine the like terms.

    $$
-ab + 0a + 0b + 2b^2
$$

  5. Simplify the expression.

    $$
-ab + 2b^2
$$
$\boxed{-ab + 2b^2}$

Part 2.3· 10 steps

Simplify

  • 2.3.1 x0⋅x3x^0 \cdot x^3

  1. Apply the rule x0=1x^0 = 1.

    $$
1 \cdot x^3
$$

  2. Multiply.

    $$
x^3
$$
$\boxed{x^3}$
    • 2.3.2 n6m4⋅n4m2n2m6⋅n4m0\frac{n^6 m^4 \cdot n^4 m^2}{n^2 m^6 \cdot n^4 m^0}

  3. Simplify the numerator using the product rule for exponents (ax⋅ay=ax+ya^x \cdot a^y = a^{x+y}).

    $$
n^{6+4} m^{4+2} = n^{10} m^6
$$

  4. Simplify the denominator using the product rule for exponents and m0=1m^0 = 1.

    $$
n^{2+4} m^{6+0} = n^6 m^6
$$

  5. Divide the simplified numerator by the simplified denominator using the quotient rule for exponents (axay=ax−y\frac{a^x}{a^y} = a^{x-y}).

    $$
\frac{n^{10} m^6}{n^6 m^6} = n^{10-6} m^{6-6} = n^4 m^0
$$

  6. Simplify using m0=1m^0 = 1.

    $$
n^4 \cdot 1 = n^4
$$
$\boxed{n^4}$
    • 2.3.3 (−5x3y2)(3x2y4)(−2x)(-5x^3y^2)(3x^2y^4)(-2x)

  7. Multiply the numerical coefficients.

    $$
(-5)(3)(-2) = 30
$$

  8. Multiply the xx terms using the product rule for exponents.

    $$
x^3 \cdot x^2 \cdot x^1 = x^{3+2+1} = x^6
$$

  9. Multiply the yy terms using the product rule for exponents.

    $$
y^2 \cdot y^4 = y^{2+4} = y^6
$$

  10. Combine the results.

    $$
30x^6y^6
$$
$\boxed{30x^6y^6}$

Part 5

Part 5.1· 4 steps

Given the expression 4x3−2x4+7x2−5x+134x^3 - 2x^4 + 7x^2 - 5x + 13.

  • 5.1.1 How many terms are in the given expression? The terms are 4x34x^3, −2x4-2x^4, 7x27x^2, −5x-5x, and 13. There are 5 terms. 5\boxed{5}

  • 5.1.2 What is the coefficient of x2x^2? The term containing x2x^2 is 7x27x^2. The coefficient of x2x^2 is 7. 7\boxed{7}

  • 5.1.3 Write down the constant value. The constant term is 13. 13\boxed{13}

  • 5.1.4 Write the expression in a descending order form of xx. Descending order means arranging terms from the highest power of xx to the lowest. The powers of xx are 4,3,2,1,04, 3, 2, 1, 0.

    −2x4+4x3+7x2−5x+13-2x^4 + 4x^3 + 7x^2 - 5x + 13

    −2x4+4x3+7x2−5x+13\boxed{-2x^4 + 4x^3 + 7x^2 - 5x + 13}

  • 5.1.5 Determine the value of the expression if x=2x = 2.

  1. Substitute x=2x = 2 into the expression.

    $$
4(2)^3 - 2(2)^4 + 7(2)^2 - 5(2) + 13
$$

  2. Calculate the powers.

    $$
4(8) - 2(16) + 7(4) - 5(2) + 13
$$

  3. Perform the multiplications.

    $$
32 - 32 + 28 - 10 + 13
$$

  4. Perform the additions and subtractions from left to right.

    $$
0 + 28 - 10 + 13
$$
$$
18 + 13 = 31
$$
$\boxed{31}$
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