Problem 70
Question
Exercises \(70-72\) will help you prepare for the material covered in the first section of the next chapter. $$\text { Multiply: } \quad(7-3 x)(-2-5 x)$$
Step-by-Step Solution
Verified Answer
So, the product of \((7-3x)(-2-5x)\) is \(15x^2 -29x -14\).
1Step 1: Multiply the First terms
Multiply the first terms in each binomial: \(7*(-2) = -14\).
2Step 2: Multiply the Outer terms
Next, multiply the outer terms: \(7*(-5x) = -35x\).
3Step 3: Multiply the Inner terms
Then, multiply the inner terms: \(-3x*(-2) = 6x\).
4Step 4: Multiply the Last terms
Finally, multiply the last terms: \(-3x*(-5x) = 15x^2\).
5Step 5: Combine the Results
Combine all the results: \(-14 - 35x + 6x + 15x^2 = -14 -29x + 15x^2\). However, it is more standard to write in descending exponent order as \(15x^2 -29x -14\).
Key Concepts
Polynomial MultiplicationFOIL MethodBinomial ProductAlgebraic Expressions
Polynomial Multiplication
Polynomial multiplication is a fundamental concept in algebra where we multiply polynomials to find a product. A polynomial is an expression consisting of variables (also known as indeterminates), coefficients, and exponents that are combined using addition, subtraction, and multiplication.
When you multiply polynomials, you're combining these expressions in a way that every term in the first polynomial gets multiplied by every term in the second polynomial. This can become particularly complex when dealing with larger polynomials.
To handle such complexity, it's beneficial to understand and apply structured methods like the FOIL method for binomials or the area method and vertical method for larger polynomials. The method chosen often depends on the complexity of the polynomials involved and the preference of the person solving the problem.
When you multiply polynomials, you're combining these expressions in a way that every term in the first polynomial gets multiplied by every term in the second polynomial. This can become particularly complex when dealing with larger polynomials.
To handle such complexity, it's beneficial to understand and apply structured methods like the FOIL method for binomials or the area method and vertical method for larger polynomials. The method chosen often depends on the complexity of the polynomials involved and the preference of the person solving the problem.
FOIL Method
The FOIL method is a technique used to multiply two binomials. The acronym FOIL stands for First, Outer, Inner, Last, referring to the position of the terms in each binomial that are multiplied together.
After multiplying, the next step is to combine like terms, which are terms that have the same variables raised to the same power. This is essential in simplifying the polynomial to its standard form, which usually means writing the terms in descending powers of any variables.
First:
Multiply the first terms in each binomial.Outer:
Multiply the terms on the outside.Inner:
Multiply the inner terms.Last:
Multiply the last terms in each binomial.After multiplying, the next step is to combine like terms, which are terms that have the same variables raised to the same power. This is essential in simplifying the polynomial to its standard form, which usually means writing the terms in descending powers of any variables.
Binomial Product
A binomial is an algebraic expression containing two terms that are separated by either a plus or a minus sign. A binomial product, thus, is the result of multiplying two binomials together.
For example, in the multiplication \(7-3x)(-2-5x)\), we are dealing with a binomial product. Each term of the first binomial is multiplied by each term of the second binomial, and the products are then combined. This results in a polynomial, which in this case is quadratic since the highest power of the variable \(x\) is 2. Properly combining the terms and arranging them in descending order by the exponent's value gives us the final expression of the binomial product.
For example, in the multiplication \(7-3x)(-2-5x)\), we are dealing with a binomial product. Each term of the first binomial is multiplied by each term of the second binomial, and the products are then combined. This results in a polynomial, which in this case is quadratic since the highest power of the variable \(x\) is 2. Properly combining the terms and arranging them in descending order by the exponent's value gives us the final expression of the binomial product.
Algebraic Expressions
Algebraic expressions are combinations of letters and numbers using the operations of addition, subtraction, multiplication, division, and exponentiation. The letters, called variables, can represent unknown values or changing quantities, while the numbers are known as constants.
When working with algebraic expressions, one must understand how to manipulate and simplify them through various algebraic operations and properties, such as the distributive property, commutative property, and associative property. These expressions form the basis of algebra and are essential for describing and solving a broad range of mathematical problems.
When working with algebraic expressions, one must understand how to manipulate and simplify them through various algebraic operations and properties, such as the distributive property, commutative property, and associative property. These expressions form the basis of algebra and are essential for describing and solving a broad range of mathematical problems.
Other exercises in this chapter
Problem 69
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