Polynomial Operations, Scientific Notation, and Exponents – Guided Practice

Q1. State the (a) degree of the polynomial and (b) the type of polynomial for: 8x3 – 9x8 + 3x2

Background

Topic: Polynomials – Degree and Classification

This question tests your understanding of how to determine the degree of a polynomial and how to classify it based on the number of terms.

Key Terms:

• Degree of a polynomial: The highest power of the variable in the polynomial.

• Monomial: A polynomial with one term.

• Binomial: A polynomial with two terms.

• Trinomial: A polynomial with three terms.

• Polynomial: An expression with more than three terms.

Step-by-Step Guidance

1. Identify each term in the polynomial: $8x^3$, $-9x^8$, $3x^2$.

2. Determine the degree by finding the highest exponent among all terms.

3. Count the number of terms to classify the polynomial (monomial, binomial, trinomial, or polynomial).

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Q2. Add: (3x2 – 7x + 8) + (4x2 + 5x – 10)

Background

Topic: Adding Polynomials

This question checks your ability to add polynomials by combining like terms.

Key Terms and Process:

• Like terms: Terms with the same variable and exponent.

• Combine like terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Write both polynomials in a horizontal format: $(3x^2 - 7x + 8) + (4x^2 + 5x - 10)$.

2. Group like terms together: $(3x^2 + 4x^2) + (-7x + 5x) + (8 - 10)$.

3. Add the coefficients of each group of like terms.

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Q3. Subtract: (5x2 – 4x + 3) – (x2 + 8x – 12)

Background

Topic: Subtracting Polynomials

This question tests your ability to subtract one polynomial from another by distributing the negative sign and combining like terms.

Key Terms and Process:

• Distribute: Apply the negative sign to each term in the second polynomial.

• Combine like terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Rewrite the subtraction as addition of the opposite: $(5x^2 - 4x + 3) + [-(x^2 + 8x - 12)]$.

2. Distribute the negative sign: $-x^2 - 8x + 12$.

3. Combine like terms: $(5x^2 - x^2) + (-4x - 8x) + (3 + 12)$.

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Q4. Simplify: (3x2 – 2x + 5) – (6x2 + 7x – 2) + (4x2 – 9)

Background

Topic: Adding and Subtracting Multiple Polynomials

This question tests your ability to combine several polynomials by grouping like terms and performing addition and subtraction.

Key Terms and Process:

• Like terms: Terms with the same variable and exponent.

• Combine like terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Write all polynomials in a single expression: $(3x^2 - 2x + 5) - (6x^2 + 7x - 2) + (4x^2 - 9)$.

2. Distribute the negative sign to the second polynomial: $-6x^2 - 7x + 2$.

3. Combine all like terms: $x^2$ terms, $x$ terms, and constants.

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Q5. Write 5,876,000,000 in Scientific Notation

Background

Topic: Scientific Notation

This question tests your ability to convert a large number into scientific notation.

Key Terms and Formula:

• Scientific Notation: A number written as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer.

Step-by-Step Guidance

1. Identify the first nonzero digit and place the decimal after it to form a number between 1 and 10.

2. Count how many places you move the decimal point to reach this position; this number is your exponent $n$.

3. Write the number as $a \times 10^n$.

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Q6. Write 0.000000000068 in Scientific Notation

Background

Topic: Scientific Notation

This question tests your ability to convert a very small number into scientific notation.

Key Terms and Formula:

• Scientific Notation: $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer (negative for small numbers).

Step-by-Step Guidance

1. Move the decimal point to the right until you have a number between 1 and 10.

2. Count the number of places you moved the decimal; this will be a negative exponent.

3. Write the number as $a \times 10^n$.

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Q7. Write 2.98 x 10-6 in Standard Notation

Background

Topic: Scientific Notation to Standard Notation

This question tests your ability to convert a number from scientific notation to standard decimal form.

Key Terms and Process:

• Standard Notation: The usual way of writing numbers (not using powers of ten).

Step-by-Step Guidance

1. Identify the exponent on 10 (here, -6).

2. Move the decimal point in 2.98 to the left by 6 places.

3. Fill in zeros as needed to the left of the digits.

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Q8. Write 7.39 x 1012 in Standard Notation

Background

Topic: Scientific Notation to Standard Notation

This question tests your ability to convert a number from scientific notation to standard decimal form.

Key Terms and Process:

• Standard Notation: The usual way of writing numbers (not using powers of ten).

Step-by-Step Guidance

1. Identify the exponent on 10 (here, 12).

2. Move the decimal point in 7.39 to the right by 12 places.

3. Fill in zeros as needed to the right of the digits.

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Q9. Multiply and write your answer in scientific and standard notation: (4 x 103)(9 x 105)

Background

Topic: Multiplying Numbers in Scientific Notation

This question tests your ability to multiply numbers in scientific notation and express the result in both scientific and standard notation.

Key Formula:

• $(a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{m+n}$

Step-by-Step Guidance

1. Multiply the coefficients: $4 \times 9$.

2. Add the exponents: $10^3 \times 10^5 = 10^{3+5}$.

3. Combine your results into scientific notation.

4. If necessary, adjust the coefficient to be between 1 and 10, and adjust the exponent accordingly.

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Q10. Multiply and write your answer in scientific and standard notation: 2.6 x 10-4 x 102

Background

Topic: Multiplying Numbers in Scientific Notation

This question tests your ability to multiply numbers in scientific notation, especially when one factor is just a power of ten.

Key Formula:

• $(a \times 10^m) \times 10^n = a \times 10^{m+n}$

Step-by-Step Guidance

1. Add the exponents: $-4 + 2$.

2. Write the result as $2.6 \times 10^{?}$, where ? is the sum from step 1.

3. Convert to standard notation by moving the decimal point as indicated by the exponent.

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Q13. Simplify: (2x)3

Background

Topic: Exponents and Powers

This question tests your understanding of how to apply exponents to a product.

Key Formula:

• $(ab)^n = a^n b^n$

Step-by-Step Guidance

1. Recognize that $(2x)^3$ means multiplying $2x$ by itself three times.

2. Apply the exponent to both the coefficient and the variable: $2^3$ and $x^3$.

3. Write the result as a single term.

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Q22. Multiply: 3x3(4x2 + 7x – 5)

Background

Topic: Multiplying a Monomial by a Polynomial

This question tests your ability to use the distributive property to multiply a monomial by each term in a polynomial.

Key Formula:

• $a(b + c + d) = ab + ac + ad$

• $x^m \times x^n = x^{m+n}$

Step-by-Step Guidance

1. Multiply $3x^3$ by each term inside the parentheses: $4x^2$, $7x$, and $-5$.

2. For each multiplication, multiply the coefficients and add the exponents of $x$.

3. Write the resulting terms as a simplified polynomial.

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