Certainly! Let's tackle each of these questions one by one.
### Question 3: Rationalizing Factor (R.F) of [tex]\(\sqrt{a} + \sqrt{b}\)[/tex]
The rationalizing factor of an expression [tex]\(\sqrt{a} + \sqrt{b}\)[/tex] is an expression that, when multiplied with [tex]\(\sqrt{a} + \sqrt{b}\)[/tex], eliminates the square roots and results in a rational number or a rational polynomial.
To rationalize [tex]\(\sqrt{a} + \sqrt{b}\)[/tex], we multiply it by its conjugate [tex]\(\sqrt{a} - \sqrt{b}\)[/tex].
Here’s why:
[tex]\[
(\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b}) = (\sqrt{a})^2 - (\sqrt{b})^2 = a - b
\][/tex]
So, the rationalizing factor of [tex]\(\sqrt{a} + \sqrt{b}\)[/tex] is [tex]\(\sqrt{a} - \sqrt{b}\)[/tex].
### Question 4: Axis on which the coordinate [tex]\((3,0)\)[/tex] lies
The coordinate [tex]\((3,0)\)[/tex] tells us that the [tex]\(x\)[/tex]-coordinate is 3 and the [tex]\(y\)[/tex]-coordinate is 0. When the [tex]\(y\)[/tex]-coordinate is 0, the point lies on the [tex]\(x\)[/tex]-axis. Therefore, the point [tex]\((3,0)\)[/tex] lies on the [tex]\(x\)[/tex]-axis.
### Question 5: Quadrants for points [tex]\(A (-3,4)\)[/tex], [tex]\(B (5,-2)\)[/tex], and [tex]\(C (-1,-4)\)[/tex]
The quadrants on a Cartesian plane are defined as follows:
- Quadrant I: Both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are positive.
- Quadrant II: [tex]\(x\)[/tex] is negative and [tex]\(y\)[/tex] is positive.
- Quadrant III: Both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are negative.
- Quadrant IV: [tex]\(x\)[/tex] is positive and [tex]\(y\)[/tex] is negative.
Given points:
- [tex]\(A (-3, 4)\)[/tex]:
- [tex]\(x = -3\)[/tex], [tex]\(y = 4\)[/tex]. Since [tex]\(x\)[/tex] is negative and [tex]\(y\)[/tex] is positive, point [tex]\(A\)[/tex] lies in Quadrant II.
- [tex]\(B (5, -2)\)[/tex]:
- [tex]\(x = 5\)[/tex], [tex]\(y = -2\)[/tex]. Since [tex]\(x\)[/tex] is positive and [tex]\(y\)[/tex] is negative, point [tex]\(B\)[/tex] lies in Quadrant IV.
- [tex]\(C (-1, -4)\)[/tex]:
- [tex]\(x = -1\)[/tex], [tex]\(y = -4\)[/tex]. Since both [tex]\(x\)[/tex] and [tex]\(y\)[/tex] are negative, point [tex]\(C\)[/tex] lies in Quadrant III.
### Summary:
- The rationalizing factor of [tex]\(\sqrt{a} + \sqrt{b}\)[/tex] is [tex]\(\sqrt{a} - \sqrt{b}\)[/tex].
- The coordinate [tex]\((3,0)\)[/tex] lies on the [tex]\(x\)[/tex]-axis.
- The points [tex]\(A (-3,4)\)[/tex], [tex]\(B (5,-2)\)[/tex], and [tex]\(C (-1,-4)\)[/tex] lie in Quadrant II, Quadrant IV, and Quadrant III, respectively.
