Examples of product in the following topics:
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- A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors.
- Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition, because of the fact that the logarithm of a product is the sum of the logarithms of the factors:
- Relate the product rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of products
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- To expand the product of $(2x+3)(x-4)$ , distribute the terms using the FOIL method.
- multiply every term of the polynomial by the monomial and then add the resulting products together.
- and we see that this equals the sum of the products of the terms, where every term of $P(x)$ is multiplied exactly once with every term of $Q(x)$.
- Since we made sure that the product of polynomials abides the same laws as if the variables were real numbers, the evaluation of a product of two polynomials in a given point will be the same as the product of the evaluations of the polynomials, i.e.
- So the roots of a product of polynomials are exactly the roots of its factors, i.e.
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- For two matrices the final position of the product is shown below:
- The number of columns in $A$ is $2$, and the number of rows in $B$ is also $2$, therefore a product exists.
- Start with producing the product for the first row, first column element.
- This figure illustrates diagrammatically the product of two matrices A and B, showing how each intersection in the product matrix corresponds to a row of A and a column of B.
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- We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
- Another way to see that this rule is true is to apply both the power and product rules, and the fact that dividing by $y$ is the same is multiplying by $y^{-1}.$ So we can write $\log_b(x/y)=\log_b(x\cdot y^{-1}) = \log_bx + \log_b(y^{-1}) = \log_bx -\log_by. $
- By applying the product, power, and quotient rules, you could write this expression as $\log_2(x^4)+\log_2(y^9)-\log_2(z^{100}) = 4\log_2x+9\log_2y-100\log_2z.$
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- When multiplying positive and negative numbers, the sign of the product is determined by the following rules:
- The product of two positive numbers is positive.The product of one positive number and one negative number is negative.
- Calculate the sum, difference, product, and quotient of negative whole numbers
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- We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
- Then we have $\log_b(x^p) = \log_b (x \cdot x \cdots x) = \log_b x + \log_b x + \cdots +\log_b x = p\log_b x.$ Since the $p$ factors of $x$ are converted to $p $ summands by the product rule formula.
- This can be written as $\log_3 (3^x) + \log_3 9 + \log_3(x^{100}) = x+2+100\log_3 x, $ using a combination of the product and power rules.
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- However, sometimes it will be more useful to write a polynomial as a product of other polynomials with smaller degree, for example to study its zeros.
- The process of rewriting a polynomial as a product is called factoring.
- Factoring is the decomposition of an algebraic object, for example an integer or a polynomial, into a product of other objects, or factors, which when multiplied together give the original.
- In all cases, a product of simpler objects than the original (smaller integers, polynomials of smaller degree) is obtained.
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- Indirect variation is used to describe the relationship between two variables when their product is constant.
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- FOIL is an acronym for the four terms of the product:
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- The product of a positive and two negatives is positive, so we can conclude that the polynomial becomes positive as it passes $x=-3$.
- The product of two positives and a negative is negative, so we can conclude that the polynomial becomes negative as it passes $x=-1$.