Chapter 4: Problem 22
In Exercises 17–24, fi nd the product. \(\left(3 x^2+x-2\right)\left(-4 x^2-2 x-1\right)\)
Short Answer
Expert verified
The product of the given polynomials is \[-12 x^4 -10 x^3 +3 x^2 +3 x +2\]
Step by step solution
01
Distribute the terms
Start by distributing each term from the first polynomial to every term in the second polynomial. This means, multiply the first term \(3 x^2\) with all terms in the second polynomial, followed by \(x\) and \(-2\). It should look like this:\n\\(3 x^2 * -4 x^2\) , \(3 x^2 * -2 x\) , \(3 x^2 * -1\), \(x * -4 x^2\) , \(x * -2 x\) , \(x * -1\), \(-2 * -4 x^2\) , \(-2 * -2 x\), \(-2 * -1\)
02
Simplify each product
Next, simplify each product from the distribution. This requires just multiplication and combination of like terms:\n\\[-12 x^4 -6 x^3 -3 x^2 -4 x^3 -2 x^2 -x +8 x^2 +4 x +2\]
03
Combine like terms
Now, combine like terms. Look for terms with the same variable and exponent and add or subtract their coefficients to simplify:\n\\[-12 x^4 -10 x^3 +3 x^2 +3 x +2\]
04
State the resultant polynomial
After all the simplifications, the resultant polynomial after multiplication is \[-12 x^4 -10 x^3 +3 x^2 +3 x +2\]. Ensure to arrange the terms in a decreasing power of x
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Distributive Property
Understanding the distributive property is key to mastering polynomial multiplication. This property is essentially a rule that allows us to multiply a single term by each term within a bracket, individually. For example, if you have \( a(b + c) \), the distributive property lets you expand this to \( ab + ac \).
When we apply this to polynomials, each term in the first polynomial is multiplied by every term in the second polynomial. This produces a set of new terms that represents the product of the polynomials. To demonstrate, let's consider the example \( (3x^2 + x - 2)(-4x^2 - 2x - 1) \). You multiply each term of the first polynomial — \(3x^2\), \(x\), and \$(-2)\$ — by each of the terms in the second polynomial \$(-4x^2)\$, \$(-2x)\$, and \$(-1)\$, which results in a series of individual products that new terms are built from.
This concept is fundamental for polynomial multiplication and must be applied accurately to achieve the correct answer. The distributive property ensures that all parts of the polynomials interact, leaving no term behind, which is crucial for reaching the subsequent step of combining like terms.
When we apply this to polynomials, each term in the first polynomial is multiplied by every term in the second polynomial. This produces a set of new terms that represents the product of the polynomials. To demonstrate, let's consider the example \( (3x^2 + x - 2)(-4x^2 - 2x - 1) \). You multiply each term of the first polynomial — \(3x^2\), \(x\), and \$(-2)\$ — by each of the terms in the second polynomial \$(-4x^2)\$, \$(-2x)\$, and \$(-1)\$, which results in a series of individual products that new terms are built from.
This concept is fundamental for polynomial multiplication and must be applied accurately to achieve the correct answer. The distributive property ensures that all parts of the polynomials interact, leaving no term behind, which is crucial for reaching the subsequent step of combining like terms.
Combining Like Terms
Once the distributive property has been applied, and you have a series of terms from the multiplication, the next stage is to combine like terms. In the context of polynomials, 'like terms' are terms that have the same variables raised to the same power. The coefficients of these terms can be added or subtracted from one another.
In our example, after distributing we have terms like \( -6x^3 \) and \( -4x^3 \) which can be combined because they're both cubic terms of \( x \). When you identify like terms, simply add or subtract the coefficients and keep the variable part unchanged. This step reduces the number of terms and simplifies the polynomial.
This is a form of simplification that is essential to managing more complex polynomial equations, and meticulous attention should be paid to ensure that only truly like terms are combined. For instance, \( -12x^4 \) cannot be combined with \( -10x^3 \) because they are not like terms; one is a quartic term while the other is a cubic term. This process is repeated until no more like terms can be found.
In our example, after distributing we have terms like \( -6x^3 \) and \( -4x^3 \) which can be combined because they're both cubic terms of \( x \). When you identify like terms, simply add or subtract the coefficients and keep the variable part unchanged. This step reduces the number of terms and simplifies the polynomial.
This is a form of simplification that is essential to managing more complex polynomial equations, and meticulous attention should be paid to ensure that only truly like terms are combined. For instance, \( -12x^4 \) cannot be combined with \( -10x^3 \) because they are not like terms; one is a quartic term while the other is a cubic term. This process is repeated until no more like terms can be found.
Polynomial Simplification
Polynomial simplification is a way to make a complex polynomial expression more manageable and easier to comprehend. After applying the distributive property and combining like terms, simplification might still involve several steps, such as rearranging the terms in descending powers and ensuring that there are no more like terms left to combine.
Looking at the sorted terms of our polynomial product, \( -12x^4 -10x^3 +3x^2 +3x +2 \) is a simplified polynomial. The simplification process aids in identifying the polynomial's degree (which is the highest exponent; in this case, 4), and provides clear structure for further mathematical operations or evaluations. It is the final step in the multiplication process that gives us a comprehensive view of the polynomial's characteristics.
It's important to always review each step carefully to avoid mistakes. Even small errors in the simplification process can lead to incorrect results. Precise simplification is therefore not only beneficial for understanding the current problem but is also an essential skill for solving more advanced mathematical problems involving polynomials.
Looking at the sorted terms of our polynomial product, \( -12x^4 -10x^3 +3x^2 +3x +2 \) is a simplified polynomial. The simplification process aids in identifying the polynomial's degree (which is the highest exponent; in this case, 4), and provides clear structure for further mathematical operations or evaluations. It is the final step in the multiplication process that gives us a comprehensive view of the polynomial's characteristics.
It's important to always review each step carefully to avoid mistakes. Even small errors in the simplification process can lead to incorrect results. Precise simplification is therefore not only beneficial for understanding the current problem but is also an essential skill for solving more advanced mathematical problems involving polynomials.
