consider the polynomial (x-a)(x-b) and the real number line
In each of the three subintervals of the line, write the sign of each factor and the sign of the product.For which values does the polynomial possibly change signs?
Well, well, well, let's dive into the world of polynomials and the real number line!
So, we have the polynomial (x-a)(x-b), and we need to analyze the signs of each factor and the product in the three subintervals of the number line.
Let's break it down, shall we?
In the interval (-∞, a), both (x-a) and (x-b) will be negative factors. Since multiplying two negatives gives us a positive product, the product of the factors will be positive in this interval.
In the interval (a, b), the factor (x-a) will become positive, while (x-b) remains negative. When we multiply a positive and a negative, it results in a negative product. Therefore, the product of the factors will be negative in this interval.
Lastly, in the interval (b, ∞), both (x-a) and (x-b) will be positive. And as two positives multiplied together will always yield a positive product, the product of the factors will be positive in this interval as well.
Now, here comes the fun part! The polynomial can possibly change signs at the values a and b, since those are the points where the factors switch from positive to negative or vice versa. So, if you want the polynomial to change signs, make sure your x value hits a or b.
I hope this explanation doesn't factor in too much confusion!
To determine the sign of each factor and the sign of the product in each subinterval of the real number line, we need to consider the values of the factors (x-a) and (x-b) in each subinterval.
Let's consider the three subintervals on the real number line:
1. Subinterval (-∞, a):
For this subinterval, let's plug in a value smaller than a into the factors. If we choose a very small negative number (e.g., -1), then (x-a) will be negative since (-1 - a) is negative. Similarly, (x-b) will be negative since (-1 - b) is also negative. Therefore, the product will be positive since a negative factor multiplied by a negative factor gives a positive product.
Sign of (x-a): Negative
Sign of (x-b): Negative
Sign of (x-a)(x-b): Positive
2. Subinterval (a, b):
For this subinterval, let's choose a value between a and b, such as the midpoint (a+b)/2. Plugging this value into the factors, we get:
(x-a) > 0 since ((a+b)/2 - a) is positive.
(x-b) < 0 since ((a+b)/2 - b) is negative.
Therefore, the product will be negative since a positive factor multiplied by a negative factor gives a negative product.
Sign of (x-a): Positive
Sign of (x-b): Negative
Sign of (x-a)(x-b): Negative
3. Subinterval (b, ∞):
For this subinterval, let's plug in a value larger than b into the factors. If we choose a very large positive number (e.g., 1,000), then (x-a) will be positive since (1,000 - a) is positive. Similarly, (x-b) will be positive since (1,000 - b) is also positive. Therefore, the product will be positive since a positive factor multiplied by a positive factor gives a positive product.
Sign of (x-a): Positive
Sign of (x-b): Positive
Sign of (x-a)(x-b): Positive
In conclusion, the polynomial (x-a)(x-b) possibly changes signs at the points a and b, since the product changes from positive to negative as x goes from the interval (a,b).
To analyze the sign changes of the polynomial (x-a)(x-b) on the real number line, we need to consider the values of 'x' in three different subintervals. We will denote these subintervals as follows:
1. Interval 1: x < a
2. Interval 2: a < x < b
3. Interval 3: x > b
Let's examine each subinterval separately and determine the sign of each factor and the product:
Interval 1: x < a
- For this interval, both (x-a) and (x-b) will be negative since 'x' is less than both 'a' and 'b'.
- Hence, the product (x-a)(x-b) will be positive because a negative number multiplied by another negative number gives a positive result.
Interval 2: a < x < b
- In this interval, (x-a) becomes positive because 'x' is greater than 'a'. However, (x-b) remains negative since 'x' is less than 'b'.
- Therefore, the product (x-a)(x-b) will be negative because a positive number multiplied by a negative number gives a negative result.
Interval 3: x > b
- For this interval, both (x-a) and (x-b) will be positive since 'x' is greater than both 'a' and 'b'.
- Consequently, the product (x-a)(x-b) will be positive because a positive number multiplied by another positive number gives a positive result.
Now, let's analyze the sign changes:
- The polynomial can possibly change signs only when the product (x-a)(x-b) equals zero. This occurs when either (x-a) or (x-b) is zero.
- From our analysis, we see that (x-a) can be zero when x = a, and (x-b) can be zero when x = b.
- Therefore, the polynomial (x-a)(x-b) possibly changes signs at the values of x = a and x = b.
To summarize:
- In Interval 1 (x < a), the product (x-a)(x-b) is positive.
- In Interval 2 (a < x < b), the product (x-a)(x-b) is negative.
- In Interval 3 (x > b), the product (x-a)(x-b) is positive.
- The polynomial (x-a)(x-b) possibly changes signs at x = a and x = b.